static char help[] = "Single-DOF oscillator formulated as a second-order system.\n";

#include <petscts.h>

typedef struct {
  PetscReal Omega;   /* natural frequency */
  PetscReal Xi;      /* damping coefficient  */
  PetscReal u0,v0;   /* initial conditions */
} UserParams;

static void Exact(PetscReal t,PetscReal omega,PetscReal xi,PetscReal u0,PetscReal v0,PetscReal *ut,PetscReal *vt)
{
  PetscReal u,v;
  if (xi < 1) {
    PetscReal a  = xi*omega;
    PetscReal w  = PetscSqrtReal(1-xi*xi)*omega;
    PetscReal C1 = (v0 + a*u0)/w;
    PetscReal C2 = u0;
    u = PetscExpReal(-a*t) * (C1*PetscSinReal(w*t) + C2*PetscCosReal(w*t));
    v = (- a * PetscExpReal(-a*t) * (C1*PetscSinReal(w*t) + C2*PetscCosReal(w*t)) + w * PetscExpReal(-a*t) * (C1*PetscCosReal(w*t) - C2*PetscSinReal(w*t)));
  } else if (xi > 1) {
    PetscReal w  = PetscSqrtReal(xi*xi-1)*omega;
    PetscReal C1 = (w*u0 + xi*u0 + v0)/(2*w);
    PetscReal C2 = (w*u0 - xi*u0 - v0)/(2*w);
    u = C1*PetscExpReal((-xi+w)*t) + C2*PetscExpReal((-xi-w)*t);
    v = C1*(-xi+w)*PetscExpReal((-xi+w)*t) + C2*(-xi-w)*PetscExpReal((-xi-w)*t);
  } else {
    PetscReal a  = xi*omega;
    PetscReal C1 = v0 + a*u0;
    PetscReal C2 = u0;
    u = (C1*t + C2) * PetscExpReal(-a*t);
    v = (C1 - a*(C1*t + C2)) * PetscExpReal(-a*t);
  }
  if (ut) *ut = u;
  if (vt) *vt = v;
}

PetscErrorCode Solution(TS ts,PetscReal t,Vec X,void *ctx)
{
  UserParams     *user = (UserParams*)ctx;
  PetscReal      u,v;
  PetscScalar    *x;

  PetscFunctionBegin;
  Exact(t,user->Omega,user->Xi,user->u0,user->v0,&u,&v);
  PetscCall(VecGetArray(X,&x));
  x[0] = (PetscScalar)u;
  PetscCall(VecRestoreArray(X,&x));
  PetscFunctionReturn(0);
}

PetscErrorCode Residual1(TS ts,PetscReal t,Vec U,Vec A,Vec R,void *ctx)
{
  UserParams        *user = (UserParams*)ctx;
  PetscReal         Omega = user->Omega;
  const PetscScalar *u,*a;
  PetscScalar       *r;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U,&u));
  PetscCall(VecGetArrayRead(A,&a));
  PetscCall(VecGetArrayWrite(R,&r));

  r[0] = a[0] + (Omega*Omega)*u[0];

  PetscCall(VecRestoreArrayRead(U,&u));
  PetscCall(VecRestoreArrayRead(A,&a));
  PetscCall(VecRestoreArrayWrite(R,&r));
  PetscCall(VecAssemblyBegin(R));
  PetscCall(VecAssemblyEnd(R));
  PetscFunctionReturn(0);
}

PetscErrorCode Tangent1(TS ts,PetscReal t,Vec U,Vec A,PetscReal shiftA,Mat J,Mat P,void *ctx)
{
  UserParams     *user = (UserParams*)ctx;
  PetscReal      Omega = user->Omega;
  PetscReal      T = 0;

  PetscFunctionBegin;

  T = shiftA + (Omega*Omega);

  PetscCall(MatSetValue(P,0,0,T,INSERT_VALUES));
  PetscCall(MatAssemblyBegin(P,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd  (P,MAT_FINAL_ASSEMBLY));
  if (J != P) {
    PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(0);
}

PetscErrorCode Residual2(TS ts,PetscReal t,Vec U,Vec V,Vec A,Vec R,void *ctx)
{
  UserParams         *user = (UserParams*)ctx;
  PetscReal          Omega = user->Omega, Xi = user->Xi;
  const PetscScalar *u,*v,*a;
  PetscScalar       *r;

  PetscFunctionBegin;
  PetscCall(VecGetArrayRead(U,&u));
  PetscCall(VecGetArrayRead(V,&v));
  PetscCall(VecGetArrayRead(A,&a));
  PetscCall(VecGetArrayWrite(R,&r));

  r[0] = a[0] + (2*Xi*Omega)*v[0] + (Omega*Omega)*u[0];

  PetscCall(VecRestoreArrayRead(U,&u));
  PetscCall(VecRestoreArrayRead(V,&v));
  PetscCall(VecRestoreArrayRead(A,&a));
  PetscCall(VecRestoreArrayWrite(R,&r));
  PetscCall(VecAssemblyBegin(R));
  PetscCall(VecAssemblyEnd(R));
  PetscFunctionReturn(0);
}

PetscErrorCode Tangent2(TS ts,PetscReal t,Vec U,Vec V,Vec A,PetscReal shiftV,PetscReal shiftA,Mat J,Mat P,void *ctx)
{
  UserParams     *user = (UserParams*)ctx;
  PetscReal      Omega = user->Omega, Xi = user->Xi;
  PetscReal      T = 0;

  PetscFunctionBegin;

  T = shiftA + shiftV * (2*Xi*Omega) + (Omega*Omega);

  PetscCall(MatSetValue(P,0,0,T,INSERT_VALUES));
  PetscCall(MatAssemblyBegin(P,MAT_FINAL_ASSEMBLY));
  PetscCall(MatAssemblyEnd  (P,MAT_FINAL_ASSEMBLY));
  if (J != P) {
    PetscCall(MatAssemblyBegin(J,MAT_FINAL_ASSEMBLY));
    PetscCall(MatAssemblyEnd(J,MAT_FINAL_ASSEMBLY));
  }
  PetscFunctionReturn(0);
}

int main(int argc, char *argv[])
{
  PetscMPIInt    size;
  TS             ts;
  Vec            R;
  Mat            J;
  Vec            U,V;
  PetscScalar    *u,*v;
  UserParams     user = {/*Omega=*/ 1, /*Xi=*/ 0, /*u0=*/ 1, /*,v0=*/ 0};

  PetscFunctionBeginUser;
  PetscCall(PetscInitialize(&argc,&argv,NULL,help));
  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_SELF,"","ex43 options","");
  PetscCall(PetscOptionsReal("-frequency","Natual frequency",__FILE__,user.Omega,&user.Omega,NULL));
  PetscCall(PetscOptionsReal("-damping","Damping coefficient",__FILE__,user.Xi,&user.Xi,NULL));
  PetscCall(PetscOptionsReal("-initial_u","Initial displacement",__FILE__,user.u0,&user.u0,NULL));
  PetscCall(PetscOptionsReal("-initial_v","Initial velocity",__FILE__,user.v0,&user.v0,NULL));
  PetscOptionsEnd();

  PetscCall(TSCreate(PETSC_COMM_SELF,&ts));
  PetscCall(TSSetType(ts,TSALPHA2));
  PetscCall(TSSetMaxTime(ts,5*(2*PETSC_PI)));
  PetscCall(TSSetExactFinalTime(ts,TS_EXACTFINALTIME_STEPOVER));
  PetscCall(TSSetTimeStep(ts,0.01));

  PetscCall(VecCreateSeq(PETSC_COMM_SELF,1,&R));
  PetscCall(VecSetUp(R));
  PetscCall(MatCreateSeqDense(PETSC_COMM_SELF,1,1,NULL,&J));
  PetscCall(MatSetUp(J));
  if (user.Xi) {
    PetscCall(TSSetI2Function(ts,R,Residual2,&user));
    PetscCall(TSSetI2Jacobian(ts,J,J,Tangent2,&user));
  } else {
    PetscCall(TSSetIFunction(ts,R,Residual1,&user));
    PetscCall(TSSetIJacobian(ts,J,J,Tangent1,&user));
  }
  PetscCall(VecDestroy(&R));
  PetscCall(MatDestroy(&J));
  PetscCall(TSSetSolutionFunction(ts,Solution,&user));

  PetscCall(VecCreateSeq(PETSC_COMM_SELF,1,&U));
  PetscCall(VecCreateSeq(PETSC_COMM_SELF,1,&V));
  PetscCall(VecGetArrayWrite(U,&u));
  PetscCall(VecGetArrayWrite(V,&v));
  u[0] = user.u0;
  v[0] = user.v0;
  PetscCall(VecRestoreArrayWrite(U,&u));
  PetscCall(VecRestoreArrayWrite(V,&v));

  PetscCall(TS2SetSolution(ts,U,V));
  PetscCall(TSSetFromOptions(ts));
  PetscCall(TSSolve(ts,NULL));

  PetscCall(VecDestroy(&U));
  PetscCall(VecDestroy(&V));
  PetscCall(TSDestroy(&ts));
  PetscCall(PetscFinalize());
  return 0;
}

/*TEST

    test:
      suffix: a
      args: -ts_max_steps 10 -ts_view
      requires: !single

    test:
      suffix: b
      args: -ts_max_steps 10 -ts_rtol 0 -ts_atol 1e-5 -ts_adapt_type basic -ts_adapt_monitor
      requires: !single

TEST*/