xref: /petsc/src/ts/tutorials/ex31.c (revision 3c633725528e547aaaa9b672a746f5d686a276e1)

static char help[] = "Solves the ordinary differential equations (IVPs) using explicit and implicit time-integration methods.\n";

/*

  Concepts:   TS
  Useful command line parameters:
  -problem <hull1972a1>: choose which problem to solve (see references
                      for complete listing of problems).
  -ts_type <euler>: specify time-integrator
  -ts_adapt_type <basic>: specify time-step adapting (none,basic,advanced)
  -refinement_levels <1>: number of refinement levels for convergence analysis
  -refinement_factor <2.0>: factor to refine time step size by for convergence analysis
  -dt <0.01>: specify time step (initial time step for convergence analysis)

*/

/*
List of cases and their names in the code:-
  From Hull, T.E., Enright, W.H., Fellen, B.M., and Sedgwick, A.E.,
      "Comparing Numerical Methods for Ordinary Differential
       Equations", SIAM J. Numer. Anal., 9(4), 1972, pp. 603 - 635
    A1 -> "hull1972a1" (exact solution available)
    A2 -> "hull1972a2" (exact solution available)
    A3 -> "hull1972a3" (exact solution available)
    A4 -> "hull1972a4" (exact solution available)
    A5 -> "hull1972a5"
    B1 -> "hull1972b1"
    B2 -> "hull1972b2"
    B3 -> "hull1972b3"
    B4 -> "hull1972b4"
    B5 -> "hull1972b5"
    C1 -> "hull1972c1"
    C2 -> "hull1972c2"
    C3 -> "hull1972c3"
    C4 -> "hull1972c4"

 From Constantinescu, E. "Estimating Global Errors in Time Stepping" ArXiv e-prints,
       https://arxiv.org/abs/1503.05166, 2016

    Kulikov2013I -> "kulik2013i"

*/

#include <petscts.h>

/* Function declarations */
PetscErrorCode (*RHSFunction) (TS,PetscReal,Vec,Vec,void*);
PetscErrorCode (*RHSJacobian) (TS,PetscReal,Vec,Mat,Mat,void*);
PetscErrorCode (*IFunction)   (TS,PetscReal,Vec,Vec,Vec,void*);
PetscErrorCode (*IJacobian)   (TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*);

/* Returns the size of the system of equations depending on problem specification */
PetscInt GetSize(const char *p)
{
  PetscFunctionBegin;
  if      ((!strcmp(p,"hull1972a1"))
         ||(!strcmp(p,"hull1972a2"))
         ||(!strcmp(p,"hull1972a3"))
         ||(!strcmp(p,"hull1972a4"))
         ||(!strcmp(p,"hull1972a5"))) PetscFunctionReturn(1);
  else if  (!strcmp(p,"hull1972b1")) PetscFunctionReturn(2);
  else if ((!strcmp(p,"hull1972b2"))
         ||(!strcmp(p,"hull1972b3"))
         ||(!strcmp(p,"hull1972b4"))
         ||(!strcmp(p,"hull1972b5"))) PetscFunctionReturn(3);
  else if ((!strcmp(p,"kulik2013i"))) PetscFunctionReturn(4);
  else if ((!strcmp(p,"hull1972c1"))
         ||(!strcmp(p,"hull1972c2"))
         ||(!strcmp(p,"hull1972c3"))) PetscFunctionReturn(10);
  else if  (!strcmp(p,"hull1972c4")) PetscFunctionReturn(51);
  else PetscFunctionReturn(-1);
}

/****************************************************************/

/* Problem specific functions */

/* Hull, 1972, Problem A1 */

PetscErrorCode RHSFunction_Hull1972A1(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Hull1972A1(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value = -1.0;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972A1(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972A1(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value = a - 1.0;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A2 */

PetscErrorCode RHSFunction_Hull1972A2(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -0.5*y[0]*y[0]*y[0];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Hull1972A2(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = -0.5*3.0*y[0]*y[0];
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972A2(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -0.5*y[0]*y[0]*y[0];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972A2(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = a + 0.5*3.0*y[0]*y[0];
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A3 */

PetscErrorCode RHSFunction_Hull1972A3(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscScalar       *f;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = y[0]*PetscCosReal(t);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Hull1972A3(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value = PetscCosReal(t);

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972A3(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = y[0]*PetscCosReal(t);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972A3(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value = a - PetscCosReal(t);

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A4 */

PetscErrorCode RHSFunction_Hull1972A4(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = (0.25*y[0])*(1.0-0.05*y[0]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Hull1972A4(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = 0.25*(1.0-0.05*y[0]) - (0.25*y[0])*0.05;
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972A4(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = (0.25*y[0])*(1.0-0.05*y[0]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972A4(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = a - 0.25*(1.0-0.05*y[0]) + (0.25*y[0])*0.05;
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A5 */

PetscErrorCode RHSFunction_Hull1972A5(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = (y[0]-t)/(y[0]+t);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Hull1972A5(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = 2*t/((t+y[0])*(t+y[0]));
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972A5(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = (y[0]-t)/(y[0]+t);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972A5(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row = 0,col = 0;
  PetscScalar       value;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value = a - 2*t/((t+y[0])*(t+y[0]));
  ierr = MatSetValues(A,1,&row,1,&col,&value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B1 */

PetscErrorCode RHSFunction_Hull1972B1(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = 2.0*(y[0] - y[0]*y[1]);
  f[1] = -(y[1]-y[0]*y[1]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972B1(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = 2.0*(y[0] - y[0]*y[1]);
  f[1] = -(y[1]-y[0]*y[1]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972B1(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[2] = {0,1};
  PetscScalar       value[2][2];

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value[0][0] = a - 2.0*(1.0-y[1]);    value[0][1] = 2.0*y[0];
  value[1][0] = -y[1];                 value[1][1] = a + 1.0 - y[0];
  ierr = MatSetValues(A,2,&row[0],2,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B2 */

PetscErrorCode RHSFunction_Hull1972B2(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0] + y[1];
  f[1] = y[0] - 2.0*y[1] + y[2];
  f[2] = y[1] - y[2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972B2(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0] + y[1];
  f[1] = y[0] - 2.0*y[1] + y[2];
  f[2] = y[1] - y[2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972B2(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[3] = {0,1,2};
  PetscScalar       value[3][3];

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value[0][0] = a + 1.0;  value[0][1] = -1.0;     value[0][2] = 0;
  value[1][0] = -1.0;     value[1][1] = a + 2.0;  value[1][2] = -1.0;
  value[2][0] = 0;        value[2][1] = -1.0;     value[2][2] = a + 1.0;
  ierr = MatSetValues(A,3,&row[0],3,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B3 */

PetscErrorCode RHSFunction_Hull1972B3(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  f[1] = y[0] - y[1]*y[1];
  f[2] = y[1]*y[1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972B3(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  f[1] = y[0] - y[1]*y[1];
  f[2] = y[1]*y[1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972B3(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[3] = {0,1,2};
  PetscScalar       value[3][3];

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value[0][0] = a + 1.0; value[0][1] = 0;             value[0][2] = 0;
  value[1][0] = -1.0;    value[1][1] = a + 2.0*y[1];  value[1][2] = 0;
  value[2][0] = 0;       value[2][1] = -2.0*y[1];     value[2][2] = a;
  ierr = MatSetValues(A,3,&row[0],3,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B4 */

PetscErrorCode RHSFunction_Hull1972B4(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[1] - y[0]*y[2]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  f[1] =  y[0] - y[1]*y[2]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  f[2] = y[0]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972B4(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[1] - y[0]*y[2]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  f[1] =  y[0] - y[1]*y[2]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  f[2] = y[0]/PetscSqrtScalar(y[0]*y[0]+y[1]*y[1]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972B4(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[3] = {0,1,2};
  PetscScalar       value[3][3],fac,fac2;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  fac  = PetscPowScalar(y[0]*y[0]+y[1]*y[1],-1.5);
  fac2 = PetscPowScalar(y[0]*y[0]+y[1]*y[1],-0.5);
  value[0][0] = a + (y[1]*y[1]*y[2])*fac;
  value[0][1] = 1.0 - (y[0]*y[1]*y[2])*fac;
  value[0][2] = y[0]*fac2;
  value[1][0] = -1.0 - y[0]*y[1]*y[2]*fac;
  value[1][1] = a + y[0]*y[0]*y[2]*fac;
  value[1][2] = y[1]*fac2;
  value[2][0] = -y[1]*y[1]*fac;
  value[2][1] = y[0]*y[1]*fac;
  value[2][2] = a;
  ierr = MatSetValues(A,3,&row[0],3,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B5 */

PetscErrorCode RHSFunction_Hull1972B5(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = y[1]*y[2];
  f[1] = -y[0]*y[2];
  f[2] = -0.51*y[0]*y[1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972B5(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = y[1]*y[2];
  f[1] = -y[0]*y[2];
  f[2] = -0.51*y[0]*y[1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972B5(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[3] = {0,1,2};
  PetscScalar       value[3][3];

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  value[0][0] = a;          value[0][1] = -y[2];      value[0][2] = -y[1];
  value[1][0] = y[2];       value[1][1] = a;          value[1][2] = y[0];
  value[2][0] = 0.51*y[1];  value[2][1] = 0.51*y[0];  value[2][2] = a;
  ierr = MatSetValues(A,3,&row[0],3,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Kulikov, 2013, Problem I */

PetscErrorCode RHSFunction_Kulikov2013I(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = 2.*t*PetscPowScalar(y[1],1./5.)*y[3];
  f[1] = 10.*t*y[3]*PetscExpScalar(5.0*(y[2]-1.));
  f[2] = 2.*t*y[3];
  f[3] = -2.*t*PetscLogScalar(y[0]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Kulikov2013I(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[4] = {0,1,2,3};
  PetscScalar       value[4][4];
  PetscScalar       m1,m2;
  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  m1=(2.*t*y[3])/(5.*PetscPowScalar(y[1],4./5.));
  m2=2.*t*PetscPowScalar(y[1],1./5.);
  value[0][0] = 0. ;        value[0][1] = m1; value[0][2] = 0.;  value[0][3] = m2;
  m1=50.*t*y[3]*PetscExpScalar(5.0*(y[2]-1.));
  m2=10.*t*PetscExpScalar(5.0*(y[2]-1.));
  value[1][0] = 0.;        value[1][1] = 0. ; value[1][2] = m1; value[1][3] = m2;
  value[2][0] = 0.;        value[2][1] = 0.;  value[2][2] = 0.; value[2][3] = 2*t;
  value[3][0] = -2.*t/y[0];value[3][1] = 0.;  value[3][2] = 0.; value[3][3] = 0.;
  ierr = MatSetValues(A,4,&row[0],4,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Kulikov2013I(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = 2.*t*PetscPowScalar(y[1],1./5.)*y[3];
  f[1] = 10.*t*y[3]*PetscExpScalar(5.0*(y[2]-1.));
  f[2] = 2.*t*y[3];
  f[3] = -2.*t*PetscLogScalar(y[0]);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Kulikov2013I(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          row[4] = {0,1,2,3};
  PetscScalar       value[4][4];
  PetscScalar       m1,m2;

  PetscFunctionBegin;
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  m1=(2.*t*y[3])/(5.*PetscPowScalar(y[1],4./5.));
  m2=2.*t*PetscPowScalar(y[1],1./5.);
  value[0][0] = a ;        value[0][1] = m1;  value[0][2] = 0.; value[0][3] = m2;
  m1=50.*t*y[3]*PetscExpScalar(5.0*(y[2]-1.));
  m2=10.*t*PetscExpScalar(5.0*(y[2]-1.));
  value[1][0] = 0.;        value[1][1] = a ;  value[1][2] = m1; value[1][3] = m2;
  value[2][0] = 0.;        value[2][1] = 0.;  value[2][2] = a;  value[2][3] = 2*t;
  value[3][0] = -2.*t/y[0];value[3][1] = 0.;  value[3][2] = 0.; value[3][3] = a;
  ierr = MatSetValues(A,4,&row[0],4,&row[0],&value[0][0],INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem C1 */

PetscErrorCode RHSFunction_Hull1972C1(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  for (i = 1; i < N-1; i++) {
    f[i] = y[i-1] - y[i];
  }
  f[N-1] = y[N-2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972C1(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  for (i = 1; i < N-1; i++) {
    f[i] = y[i-1] - y[i];
  }
  f[N-1] = y[N-2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972C1(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          N,i,col[2];
  PetscScalar       value[2];

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  i = 0;
  value[0] = a+1; col[0] = 0;
  value[1] =  0;  col[1] = 1;
  ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  for (i = 0; i < N; i++) {
    value[0] =  -1; col[0] = i-1;
    value[1] = a+1; col[1] = i;
    ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  }
  i = N-1;
  value[0] = -1;  col[0] = N-2;
  value[1] = a;   col[1] = N-1;
  ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem C2 */

PetscErrorCode RHSFunction_Hull1972C2(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscScalar       *f;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  for (i = 1; i < N-1; i++) {
    f[i] = (PetscReal)i*y[i-1] - (PetscReal)(i+1)*y[i];
  }
  f[N-1] = (PetscReal)(N-1)*y[N-2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972C2(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -y[0];
  for (i = 1; i < N-1; i++) {
    f[i] = (PetscReal)i*y[i-1] - (PetscReal)(i+1)*y[i];
  }
  f[N-1] = (PetscReal)(N-1)*y[N-2];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972C2(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscInt          N,i,col[2];
  PetscScalar       value[2];

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  i = 0;
  value[0] = a+1;                 col[0] = 0;
  value[1] = 0;                   col[1] = 1;
  ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  for (i = 0; i < N; i++) {
    value[0] = -(PetscReal) i;      col[0] = i-1;
    value[1] = a+(PetscReal)(i+1);  col[1] = i;
    ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  }
  i = N-1;
  value[0] = -(PetscReal) (N-1);  col[0] = N-2;
  value[1] = a;                   col[1] = N-1;
  ierr = MatSetValues(A,1,&i,2,col,value,INSERT_VALUES);CHKERRQ(ierr);
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem C3 and C4 */

PetscErrorCode RHSFunction_Hull1972C34(TS ts, PetscReal t, Vec Y, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -2.0*y[0] + y[1];
  for (i = 1; i < N-1; i++) {
    f[i] = y[i-1] - 2.0*y[i] + y[i+1];
  }
  f[N-1] = y[N-2] - 2.0*y[N-1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IFunction_Hull1972C34(TS ts, PetscReal t, Vec Y, Vec Ydot, Vec F, void *s)
{
  PetscErrorCode    ierr;
  PetscScalar       *f;
  const PetscScalar *y;
  PetscInt          N,i;

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecGetArray(F,&f);CHKERRQ(ierr);
  f[0] = -2.0*y[0] + y[1];
  for (i = 1; i < N-1; i++) {
    f[i] = y[i-1] - 2.0*y[i] + y[i+1];
  }
  f[N-1] = y[N-2] - 2.0*y[N-1];
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  ierr = VecRestoreArray(F,&f);CHKERRQ(ierr);
  /* Left hand side = ydot - f(y) */
  ierr = VecAYPX(F,-1.0,Ydot);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

PetscErrorCode IJacobian_Hull1972C34(TS ts, PetscReal t, Vec Y, Vec Ydot, PetscReal a, Mat A, Mat B, void *s)
{
  PetscErrorCode    ierr;
  const PetscScalar *y;
  PetscScalar       value[3];
  PetscInt          N,i,col[3];

  PetscFunctionBegin;
  ierr = VecGetSize (Y,&N);CHKERRQ(ierr);
  ierr = VecGetArrayRead(Y,&y);CHKERRQ(ierr);
  for (i = 0; i < N; i++) {
    if (i == 0) {
      value[0] = a+2;  col[0] = i;
      value[1] =  -1;  col[1] = i+1;
      value[2] =  0;   col[2] = i+2;
    } else if (i == N-1) {
      value[0] =  0;   col[0] = i-2;
      value[1] =  -1;  col[1] = i-1;
      value[2] = a+2;  col[2] = i;
    } else {
      value[0] = -1;   col[0] = i-1;
      value[1] = a+2;  col[1] = i;
      value[2] = -1;   col[2] = i+1;
    }
    ierr = MatSetValues(A,1,&i,3,col,value,INSERT_VALUES);CHKERRQ(ierr);
  }
  ierr = MatAssemblyBegin(A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = MatAssemblyEnd  (A,MAT_FINAL_ASSEMBLY);CHKERRQ(ierr);
  ierr = VecRestoreArrayRead(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/***************************************************************************/

/* Sets the initial solution for the IVP and sets up the function pointers*/
PetscErrorCode Initialize(Vec Y, void* s)
{
  PetscErrorCode ierr;
  char          *p = (char*) s;
  PetscScalar   *y;
  PetscReal     t0;
  PetscInt      N = GetSize((const char *)s);
  PetscBool     flg;

  PetscFunctionBegin;
  VecZeroEntries(Y);
  ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
  if (!strcmp(p,"hull1972a1")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972A1;
    RHSJacobian = RHSJacobian_Hull1972A1;
    IFunction   = IFunction_Hull1972A1;
    IJacobian   = IJacobian_Hull1972A1;
  } else if (!strcmp(p,"hull1972a2")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972A2;
    RHSJacobian = RHSJacobian_Hull1972A2;
    IFunction   = IFunction_Hull1972A2;
    IJacobian   = IJacobian_Hull1972A2;
  } else if (!strcmp(p,"hull1972a3")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972A3;
    RHSJacobian = RHSJacobian_Hull1972A3;
    IFunction   = IFunction_Hull1972A3;
    IJacobian   = IJacobian_Hull1972A3;
  } else if (!strcmp(p,"hull1972a4")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972A4;
    RHSJacobian = RHSJacobian_Hull1972A4;
    IFunction   = IFunction_Hull1972A4;
    IJacobian   = IJacobian_Hull1972A4;
  } else if (!strcmp(p,"hull1972a5")) {
    y[0] = 4.0;
    RHSFunction = RHSFunction_Hull1972A5;
    RHSJacobian = RHSJacobian_Hull1972A5;
    IFunction   = IFunction_Hull1972A5;
    IJacobian   = IJacobian_Hull1972A5;
  } else if (!strcmp(p,"hull1972b1")) {
    y[0] = 1.0;
    y[1] = 3.0;
    RHSFunction = RHSFunction_Hull1972B1;
    IFunction   = IFunction_Hull1972B1;
    IJacobian   = IJacobian_Hull1972B1;
  } else if (!strcmp(p,"hull1972b2")) {
    y[0] = 2.0;
    y[1] = 0.0;
    y[2] = 1.0;
    RHSFunction = RHSFunction_Hull1972B2;
    IFunction   = IFunction_Hull1972B2;
    IJacobian   = IJacobian_Hull1972B2;
  } else if (!strcmp(p,"hull1972b3")) {
    y[0] = 1.0;
    y[1] = 0.0;
    y[2] = 0.0;
    RHSFunction = RHSFunction_Hull1972B3;
    IFunction   = IFunction_Hull1972B3;
    IJacobian   = IJacobian_Hull1972B3;
  } else if (!strcmp(p,"hull1972b4")) {
    y[0] = 3.0;
    y[1] = 0.0;
    y[2] = 0.0;
    RHSFunction = RHSFunction_Hull1972B4;
    IFunction   = IFunction_Hull1972B4;
    IJacobian   = IJacobian_Hull1972B4;
  } else if (!strcmp(p,"hull1972b5")) {
    y[0] = 0.0;
    y[1] = 1.0;
    y[2] = 1.0;
    RHSFunction = RHSFunction_Hull1972B5;
    IFunction   = IFunction_Hull1972B5;
    IJacobian   = IJacobian_Hull1972B5;
  } else if (!strcmp(p,"kulik2013i")) {
    t0=0.;
    y[0] = PetscExpReal(PetscSinReal(t0*t0));
    y[1] = PetscExpReal(5.*PetscSinReal(t0*t0));
    y[2] = PetscSinReal(t0*t0)+1.0;
    y[3] = PetscCosReal(t0*t0);
    RHSFunction = RHSFunction_Kulikov2013I;
    RHSJacobian = RHSJacobian_Kulikov2013I;
    IFunction   = IFunction_Kulikov2013I;
    IJacobian   = IJacobian_Kulikov2013I;
  } else if (!strcmp(p,"hull1972c1")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972C1;
    IFunction   = IFunction_Hull1972C1;
    IJacobian   = IJacobian_Hull1972C1;
  } else if (!strcmp(p,"hull1972c2")) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972C2;
    IFunction   = IFunction_Hull1972C2;
    IJacobian   = IJacobian_Hull1972C2;
  } else if ((!strcmp(p,"hull1972c3"))
           ||(!strcmp(p,"hull1972c4"))) {
    y[0] = 1.0;
    RHSFunction = RHSFunction_Hull1972C34;
    IFunction   = IFunction_Hull1972C34;
    IJacobian   = IJacobian_Hull1972C34;
  }
  ierr = PetscOptionsGetScalarArray(NULL,NULL,"-yinit",y,&N,&flg);CHKERRQ(ierr);
  PetscCheck((N == GetSize((const char*)s)) || !flg,PETSC_COMM_WORLD,PETSC_ERR_ARG_SIZ,"Number of initial values %D does not match problem size %D.",N,GetSize((const char*)s));
  ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  PetscFunctionReturn(0);
}

/* Calculates the exact solution to problems that have one */
PetscErrorCode ExactSolution(Vec Y, void* s, PetscReal t, PetscBool *flag)
{
  PetscErrorCode ierr;
  char          *p = (char*) s;
  PetscScalar   *y;

  PetscFunctionBegin;
  if (!strcmp(p,"hull1972a1")) {
    ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
    y[0] = PetscExpReal(-t);
    *flag = PETSC_TRUE;
    ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  } else if (!strcmp(p,"hull1972a2")) {
    ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
    y[0] = 1.0/PetscSqrtReal(t+1);
    *flag = PETSC_TRUE;
    ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  } else if (!strcmp(p,"hull1972a3")) {
    ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
    y[0] = PetscExpReal(PetscSinReal(t));
    *flag = PETSC_TRUE;
    ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  } else if (!strcmp(p,"hull1972a4")) {
    ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
    y[0] = 20.0/(1+19.0*PetscExpReal(-t/4.0));
    *flag = PETSC_TRUE;
    ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  } else if (!strcmp(p,"kulik2013i")) {
    ierr = VecGetArray(Y,&y);CHKERRQ(ierr);
    y[0] = PetscExpReal(PetscSinReal(t*t));
    y[1] = PetscExpReal(5.*PetscSinReal(t*t));
    y[2] = PetscSinReal(t*t)+1.0;
    y[3] = PetscCosReal(t*t);
    *flag = PETSC_TRUE;
    ierr = VecRestoreArray(Y,&y);CHKERRQ(ierr);
  } else {
    ierr = VecSet(Y,0);CHKERRQ(ierr);
    *flag = PETSC_FALSE;
  }
  PetscFunctionReturn(0);
}

/* Solves the specified ODE and computes the error if exact solution is available */
PetscErrorCode SolveODE(char* ptype, PetscReal dt, PetscReal tfinal, PetscInt maxiter, PetscReal *error, PetscBool *exact_flag)
{
  PetscErrorCode  ierr;             /* Error code                             */
  TS              ts;               /* time-integrator                        */
  Vec             Y;                /* Solution vector                        */
  Vec             Yex;              /* Exact solution                         */
  PetscInt        N;                /* Size of the system of equations        */
  TSType          time_scheme;      /* Type of time-integration scheme        */
  Mat             Jac = NULL;       /* Jacobian matrix                        */
  Vec             Yerr;             /* Auxiliary solution vector              */
  PetscReal       err_norm;         /* Estimated error norm                   */
  PetscReal       final_time;       /* Actual final time from the integrator  */

  PetscFunctionBegin;
  N = GetSize((const char *)&ptype[0]);
  PetscCheck(N >= 0,PETSC_COMM_WORLD,PETSC_ERR_ARG_SIZ,"Illegal problem specification.");
  ierr = VecCreate(PETSC_COMM_WORLD,&Y);CHKERRQ(ierr);
  ierr = VecSetSizes(Y,N,PETSC_DECIDE);CHKERRQ(ierr);
  ierr = VecSetUp(Y);CHKERRQ(ierr);
  ierr = VecSet(Y,0);CHKERRQ(ierr);

  /* Initialize the problem */
  ierr = Initialize(Y,&ptype[0]);CHKERRQ(ierr);

  /* Create and initialize the time-integrator                            */
  ierr = TSCreate(PETSC_COMM_WORLD,&ts);CHKERRQ(ierr);
  /* Default time integration options                                     */
  ierr = TSSetType(ts,TSRK);CHKERRQ(ierr);
  ierr = TSSetMaxSteps(ts,maxiter);CHKERRQ(ierr);
  ierr = TSSetMaxTime(ts,tfinal);CHKERRQ(ierr);
  ierr = TSSetTimeStep(ts,dt);CHKERRQ(ierr);
  ierr = TSSetExactFinalTime(ts,TS_EXACTFINALTIME_MATCHSTEP);CHKERRQ(ierr);
  /* Read command line options for time integration                       */
  ierr = TSSetFromOptions(ts);CHKERRQ(ierr);
  /* Set solution vector                                                  */
  ierr = TSSetSolution(ts,Y);CHKERRQ(ierr);
  /* Specify left/right-hand side functions                               */
  ierr = TSGetType(ts,&time_scheme);CHKERRQ(ierr);

  if ((!strcmp(time_scheme,TSEULER)) || (!strcmp(time_scheme,TSRK)) || (!strcmp(time_scheme,TSSSP) || (!strcmp(time_scheme,TSGLEE)))) {
    /* Explicit time-integration -> specify right-hand side function ydot = f(y) */
    ierr = TSSetRHSFunction(ts,NULL,RHSFunction,&ptype[0]);CHKERRQ(ierr);
    ierr = MatCreate(PETSC_COMM_WORLD,&Jac);CHKERRQ(ierr);
    ierr = MatSetSizes(Jac,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
    ierr = MatSetFromOptions(Jac);CHKERRQ(ierr);
    ierr = MatSetUp(Jac);CHKERRQ(ierr);
    ierr = TSSetRHSJacobian(ts,Jac,Jac,RHSJacobian,&ptype[0]);CHKERRQ(ierr);
  } else if ((!strcmp(time_scheme,TSTHETA)) || (!strcmp(time_scheme,TSBEULER)) || (!strcmp(time_scheme,TSCN)) || (!strcmp(time_scheme,TSALPHA)) || (!strcmp(time_scheme,TSARKIMEX))) {
    /* Implicit time-integration -> specify left-hand side function ydot-f(y) = 0 */
    /* and its Jacobian function                                                 */
    ierr = TSSetIFunction(ts,NULL,IFunction,&ptype[0]);CHKERRQ(ierr);
    ierr = MatCreate(PETSC_COMM_WORLD,&Jac);CHKERRQ(ierr);
    ierr = MatSetSizes(Jac,PETSC_DECIDE,PETSC_DECIDE,N,N);CHKERRQ(ierr);
    ierr = MatSetFromOptions(Jac);CHKERRQ(ierr);
    ierr = MatSetUp(Jac);CHKERRQ(ierr);
    ierr = TSSetIJacobian(ts,Jac,Jac,IJacobian,&ptype[0]);CHKERRQ(ierr);
  }

  /* Solve */
  ierr = TSSolve(ts,Y);CHKERRQ(ierr);
  ierr = TSGetTime(ts,&final_time);CHKERRQ(ierr);

  /* Get the estimated error, if available */
  ierr = VecDuplicate(Y,&Yerr);CHKERRQ(ierr);
  ierr = VecZeroEntries(Yerr);CHKERRQ(ierr);
  ierr = TSGetTimeError(ts,0,&Yerr);CHKERRQ(ierr);
  ierr = VecNorm(Yerr,NORM_2,&err_norm);CHKERRQ(ierr);
  ierr = VecDestroy(&Yerr);CHKERRQ(ierr);
  ierr = PetscPrintf(PETSC_COMM_WORLD,"Estimated Error = %E.\n",err_norm);CHKERRQ(ierr);

  /* Exact solution */
  ierr = VecDuplicate(Y,&Yex);CHKERRQ(ierr);
  if (PetscAbsScalar(final_time-tfinal)>2.*PETSC_MACHINE_EPSILON) {
    ierr = PetscPrintf(PETSC_COMM_WORLD,"Note: There is a difference between the prescribed final time %g and the actual final time, %g.\n",(double)tfinal,(double)final_time);CHKERRQ(ierr);
  }
  ierr = ExactSolution(Yex,&ptype[0],final_time,exact_flag);CHKERRQ(ierr);

  /* Calculate Error */
  ierr = VecAYPX(Yex,-1.0,Y);CHKERRQ(ierr);
  ierr = VecNorm(Yex,NORM_2,error);CHKERRQ(ierr);
  *error = PetscSqrtReal(((*error)*(*error))/N);

  /* Clean up and finalize */
  ierr = MatDestroy(&Jac);CHKERRQ(ierr);
  ierr = TSDestroy(&ts);CHKERRQ(ierr);
  ierr = VecDestroy(&Yex);CHKERRQ(ierr);
  ierr = VecDestroy(&Y);CHKERRQ(ierr);

  PetscFunctionReturn(0);
}

int main(int argc, char **argv)
{
  PetscErrorCode  ierr;                       /* Error code                                           */
  char            ptype[256] = "hull1972a1";  /* Problem specification                                */
  PetscInt        n_refine   = 1;             /* Number of refinement levels for convergence analysis */
  PetscReal       refine_fac = 2.0;           /* Refinement factor for dt                             */
  PetscReal       dt_initial = 0.01;          /* Initial default value of dt                          */
  PetscReal       dt;
  PetscReal       tfinal     = 20.0;          /* Final time for the time-integration                  */
  PetscInt        maxiter    = 100000;        /* Maximum number of time-integration iterations        */
  PetscReal       *error;                     /* Array to store the errors for convergence analysis   */
  PetscMPIInt     size;                      /* No of processors                                     */
  PetscBool       flag;                       /* Flag denoting availability of exact solution         */
  PetscInt        r;

  /* Initialize program */
  ierr = PetscInitialize(&argc,&argv,(char*)0,help);if (ierr) return ierr;

  /* Check if running with only 1 proc */
  ierr = MPI_Comm_size(PETSC_COMM_WORLD,&size);CHKERRMPI(ierr);
  PetscCheck(size == 1,PETSC_COMM_WORLD,PETSC_ERR_WRONG_MPI_SIZE,"Only for sequential runs");

  ierr = PetscOptionsBegin(PETSC_COMM_WORLD,NULL,"ex31",NULL);CHKERRQ(ierr);
  ierr = PetscOptionsString("-problem","Problem specification","<hull1972a1>",ptype,ptype,sizeof(ptype),NULL);CHKERRQ(ierr);
  ierr = PetscOptionsInt("-refinement_levels","Number of refinement levels for convergence analysis","<1>",n_refine,&n_refine,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-refinement_factor","Refinement factor for dt","<2.0>",refine_fac,&refine_fac,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-dt","Time step size (for convergence analysis, initial time step)","<0.01>",dt_initial,&dt_initial,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsReal("-final_time","Final time for the time-integration","<20.0>",tfinal,&tfinal,NULL);CHKERRQ(ierr);
  ierr = PetscOptionsEnd();CHKERRQ(ierr);

  ierr = PetscMalloc1(n_refine,&error);CHKERRQ(ierr);
  for (r = 0,dt = dt_initial; r < n_refine; r++) {
    error[r] = 0;
    if (r > 0) dt /= refine_fac;

    ierr = PetscPrintf(PETSC_COMM_WORLD,"Solving ODE \"%s\" with dt %f, final time %f and system size %D.\n",ptype,(double)dt,(double)tfinal,GetSize(&ptype[0]));CHKERRQ(ierr);
    ierr = SolveODE(&ptype[0],dt,tfinal,maxiter,&error[r],&flag);CHKERRQ(ierr);
    if (flag) {
      /* If exact solution available for the specified ODE */
      if (r > 0) {
        PetscReal conv_rate = (PetscLogReal(error[r]) - PetscLogReal(error[r-1])) / (-PetscLogReal(refine_fac));
        ierr = PetscPrintf(PETSC_COMM_WORLD,"Error           = %E,\tConvergence rate = %f.\n",(double)error[r],(double)conv_rate);CHKERRQ(ierr);
      } else {
        ierr = PetscPrintf(PETSC_COMM_WORLD,"Error           = %E.\n",error[r]);CHKERRQ(ierr);
      }
    }
  }
  ierr = PetscFree(error);CHKERRQ(ierr);
  ierr = PetscFinalize();
  return ierr;
}

/*TEST

    test:
      suffix: 2
      args: -ts_type glee -final_time 5 -ts_adapt_type none
      timeoutfactor: 3
      requires: !single

    test:
      suffix: 3
      args: -ts_type glee -final_time 5 -ts_adapt_type glee -ts_adapt_monitor  -ts_max_steps 50  -problem hull1972a3 -ts_adapt_glee_use_local 1
      timeoutfactor: 3
      requires: !single

    test:
      suffix: 4
      args: -ts_type glee -final_time 5 -ts_adapt_type glee -ts_adapt_monitor  -ts_max_steps 50  -problem hull1972a3  -ts_max_reject 100 -ts_adapt_glee_use_local 0
      timeoutfactor: 3
      requires: !single !__float128

TEST*/