xref: /petsc/src/ts/tutorials/ex31.c (revision 9371c9d470a9602b6d10a8bf50c9b2280a79e45a)

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

/*

  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) {
  PetscFunctionBeginUser;
  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) {
  PetscScalar       *f;
  const PetscScalar *y;

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(MatSetValues(A, 1, &row, 1, &col, &value, INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(MatSetValues(A, 1, &row, 1, &col, &value, INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A2 */

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

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

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

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

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

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

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

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

/* Hull, 1972, Problem A3 */

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(MatSetValues(A, 1, &row, 1, &col, &value, INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(MatSetValues(A, 1, &row, 1, &col, &value, INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem A4 */

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

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

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

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

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

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

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

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

/* Hull, 1972, Problem A5 */

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

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

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

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

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

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

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

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

/* Hull, 1972, Problem B1 */

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

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

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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];
  PetscCall(MatSetValues(A, 2, &row[0], 2, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B2 */

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

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

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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;
  PetscCall(MatSetValues(A, 3, &row[0], 3, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B3 */

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

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

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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;
  PetscCall(MatSetValues(A, 3, &row[0], 3, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B4 */

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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]);
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(0);
}

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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;
  PetscCall(MatSetValues(A, 3, &row[0], 3, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem B5 */

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

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

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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;
  PetscCall(MatSetValues(A, 3, &row[0], 3, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Kulikov, 2013, Problem I */

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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]);
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(0);
}

PetscErrorCode RHSJacobian_Kulikov2013I(TS ts, PetscReal t, Vec Y, Mat A, Mat B, void *s) {
  const PetscScalar *y;
  PetscInt           row[4] = {0, 1, 2, 3};
  PetscScalar        value[4][4];
  PetscScalar        m1, m2;
  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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.;
  PetscCall(MatSetValues(A, 4, &row[0], 4, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetArrayRead(Y, &y));
  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;
  PetscCall(MatSetValues(A, 4, &row[0], 4, &row[0], &value[0][0], INSERT_VALUES));
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

/* Hull, 1972, Problem C1 */

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

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

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

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

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

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

/* Hull, 1972, Problem C2 */

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

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(Y, &N));
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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];
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(Y, &N));
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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];
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  /* Left hand side = ydot - f(y) */
  PetscCall(VecAYPX(F, -1.0, Ydot));
  PetscFunctionReturn(0);
}

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

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

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

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

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(Y, &N));
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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];
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(Y, &N));
  PetscCall(VecGetArrayRead(Y, &y));
  PetscCall(VecGetArray(F, &f));
  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];
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscCall(VecRestoreArray(F, &f));
  /* Left hand side = ydot - f(y) */
  PetscCall(VecAYPX(F, -1.0, Ydot));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  PetscCall(VecGetSize(Y, &N));
  PetscCall(VecGetArrayRead(Y, &y));
  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;
    }
    PetscCall(MatSetValues(A, 1, &i, 3, col, value, INSERT_VALUES));
  }
  PetscCall(MatAssemblyBegin(A, MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd(A, MAT_FINAL_ASSEMBLY));
  PetscCall(VecRestoreArrayRead(Y, &y));
  PetscFunctionReturn(0);
}

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

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

  PetscFunctionBeginUser;
  VecZeroEntries(Y);
  PetscCall(VecGetArray(Y, &y));
  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;
  }
  PetscCall(PetscOptionsGetScalarArray(NULL, NULL, "-yinit", y, &N, &flg));
  PetscCheck((N == GetSize((const char *)s)) || !flg, PETSC_COMM_WORLD, PETSC_ERR_ARG_SIZ, "Number of initial values %" PetscInt_FMT " does not match problem size %" PetscInt_FMT ".", N, GetSize((const char *)s));
  PetscCall(VecRestoreArray(Y, &y));
  PetscFunctionReturn(0);
}

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

  PetscFunctionBeginUser;
  if (!strcmp(p, "hull1972a1")) {
    PetscCall(VecGetArray(Y, &y));
    y[0]  = PetscExpReal(-t);
    *flag = PETSC_TRUE;
    PetscCall(VecRestoreArray(Y, &y));
  } else if (!strcmp(p, "hull1972a2")) {
    PetscCall(VecGetArray(Y, &y));
    y[0]  = 1.0 / PetscSqrtReal(t + 1);
    *flag = PETSC_TRUE;
    PetscCall(VecRestoreArray(Y, &y));
  } else if (!strcmp(p, "hull1972a3")) {
    PetscCall(VecGetArray(Y, &y));
    y[0]  = PetscExpReal(PetscSinReal(t));
    *flag = PETSC_TRUE;
    PetscCall(VecRestoreArray(Y, &y));
  } else if (!strcmp(p, "hull1972a4")) {
    PetscCall(VecGetArray(Y, &y));
    y[0]  = 20.0 / (1 + 19.0 * PetscExpReal(-t / 4.0));
    *flag = PETSC_TRUE;
    PetscCall(VecRestoreArray(Y, &y));
  } else if (!strcmp(p, "kulik2013i")) {
    PetscCall(VecGetArray(Y, &y));
    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;
    PetscCall(VecRestoreArray(Y, &y));
  } else {
    PetscCall(VecSet(Y, 0));
    *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) {
  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  */

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

  /* Initialize the problem */
  PetscCall(Initialize(Y, &ptype[0]));

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

  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) */
    PetscCall(TSSetRHSFunction(ts, NULL, RHSFunction, &ptype[0]));
    PetscCall(MatCreate(PETSC_COMM_WORLD, &Jac));
    PetscCall(MatSetSizes(Jac, PETSC_DECIDE, PETSC_DECIDE, N, N));
    PetscCall(MatSetFromOptions(Jac));
    PetscCall(MatSetUp(Jac));
    PetscCall(TSSetRHSJacobian(ts, Jac, Jac, RHSJacobian, &ptype[0]));
  } 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                                                 */
    PetscCall(TSSetIFunction(ts, NULL, IFunction, &ptype[0]));
    PetscCall(MatCreate(PETSC_COMM_WORLD, &Jac));
    PetscCall(MatSetSizes(Jac, PETSC_DECIDE, PETSC_DECIDE, N, N));
    PetscCall(MatSetFromOptions(Jac));
    PetscCall(MatSetUp(Jac));
    PetscCall(TSSetIJacobian(ts, Jac, Jac, IJacobian, &ptype[0]));
  }

  /* Solve */
  PetscCall(TSSolve(ts, Y));
  PetscCall(TSGetTime(ts, &final_time));

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

  /* Exact solution */
  PetscCall(VecDuplicate(Y, &Yex));
  if (PetscAbsReal(final_time - tfinal) > 2. * PETSC_MACHINE_EPSILON) {
    PetscCall(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));
  }
  PetscCall(ExactSolution(Yex, &ptype[0], final_time, exact_flag));

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

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

  PetscFunctionReturn(0);
}

int main(int argc, char **argv) {
  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 */
  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc, &argv, (char *)0, help));

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

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

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

    PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Solving ODE \"%s\" with dt %f, final time %f and system size %" PetscInt_FMT ".\n", ptype, (double)dt, (double)tfinal, GetSize(&ptype[0])));
    PetscCall(SolveODE(&ptype[0], dt, tfinal, maxiter, &error[r], &flag));
    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));
        PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Error           = %E,\tConvergence rate = %f.\n", (double)error[r], (double)conv_rate));
      } else {
        PetscCall(PetscPrintf(PETSC_COMM_WORLD, "Error           = %E.\n", (double)error[r]));
      }
    }
  }
  PetscCall(PetscFree(error));
  PetscCall(PetscFinalize());
  return 0;
}

/*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*/