TS: Scalable ODE and DAE Solvers#

The TS library provides a framework for the scalable solution of ODEs and DAEs arising from the discretization of time-dependent PDEs.

Simple Example: Consider the PDE

\[u_t = u_{xx} \]

discretized with centered finite differences in space yielding the semi-discrete equation

\[\begin{aligned} (u_i)_t & = & \frac{u_{i+1} - 2 u_{i} + u_{i-1}}{h^2}, \\ u_t & = & \tilde{A} u;\end{aligned}\]

or with piecewise linear finite elements approximation in space \(u(x,t) \doteq \sum_i \xi_i(t) \phi_i(x)\) yielding the semi-discrete equation

\[B {\xi}'(t) = A \xi(t) \]

Now applying the backward Euler method results in

\[( B - dt^n A ) u^{n+1} = B u^n, \]

in which

\[{u^n}_i = \xi_i(t_n) \doteq u(x_i,t_n), \]
\[{\xi}'(t_{n+1}) \doteq \frac{{u^{n+1}}_i - {u^{n}}_i }{dt^{n}}, \]

\(A\) is the stiffness matrix, and \(B\) is the identity for finite differences or the mass matrix for the finite element method.

The PETSc interface for solving time dependent problems assumes the problem is written in the form

\[F(t,u,\dot{u}) = G(t,u), \quad u(t_0) = u_0. \]

In general, this is a differential algebraic equation (DAE) 4. For ODE with nontrivial mass matrices such as arise in FEM, the implicit/DAE interface significantly reduces overhead to prepare the system for algebraic solvers (SNES/KSP) by having the user assemble the correctly shifted matrix. Therefore this interface is also useful for ODE systems.

To solve an ODE or DAE one uses:

  • Function \(F(t,u,\dot{u})\)

    TSSetIFunction(TS ts,Vec R,PetscErrorCode (*f)(TS,PetscReal,Vec,Vec,Vec,void*),void *funP);

    The vector R is an optional location to store the residual. The arguments to the function f() are the timestep context, current time, input state \(u\), input time derivative \(\dot{u}\), and the (optional) user-provided context funP. If \(F(t,u,\dot{u}) = \dot{u}\) then one need not call this function.

  • Function \(G(t,u)\), if it is nonzero, is provided with the function

    TSSetRHSFunction(TS ts,Vec R,PetscErrorCode (*f)(TS,PetscReal,Vec,Vec,void*),void *funP);
  • Jacobian \(\sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n)\)
    If using a fully implicit or semi-implicit (IMEX) method one also can provide an appropriate (approximate) Jacobian matrix of \(F()\).
    TSSetIJacobian(TS ts,Mat A,Mat B,PetscErrorCode (*fjac)(TS,PetscReal,Vec,Vec,PetscReal,Mat,Mat,void*),void *jacP);

    The arguments for the function fjac() are the timestep context, current time, input state \(u\), input derivative \(\dot{u}\), input shift \(\sigma\), matrix \(A\), preconditioning matrix \(B\), and the (optional) user-provided context jacP.

    The Jacobian needed for the nonlinear system is, by the chain rule,

    \[\begin{aligned} \frac{d F}{d u^n} & = & \frac{\partial F}{\partial \dot{u}}|_{u^n} \frac{\partial \dot{u}}{\partial u}|_{u^n} + \frac{\partial F}{\partial u}|_{u^n}.\end{aligned}\]

    For any ODE integration method the approximation of \(\dot{u}\) is linear in \(u^n\) hence \(\frac{\partial \dot{u}}{\partial u}|_{u^n} = \sigma\), where the shift \(\sigma\) depends on the ODE integrator and time step but not on the function being integrated. Thus

    \[\begin{aligned} \frac{d F}{d u^n} & = & \sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n).\end{aligned}\]

    This explains why the user provide Jacobian is in the given form for all integration methods. An equivalent way to derive the formula is to note that

    \[F(t^n,u^n,\dot{u}^n) = F(t^n,u^n,w+\sigma*u^n) \]

    where \(w\) is some linear combination of previous time solutions of \(u\) so that

    \[\frac{d F}{d u^n} = \sigma F_{\dot{u}}(t^n,u^n,\dot{u}^n) + F_u(t^n,u^n,\dot{u}^n) \]

    again by the chain rule.

    For example, consider backward Euler’s method applied to the ODE \(F(t, u, \dot{u}) = \dot{u} - f(t, u)\) with \(\dot{u} = (u^n - u^{n-1})/\delta t\) and \(\frac{\partial \dot{u}}{\partial u}|_{u^n} = 1/\delta t\) resulting in

    \[\begin{aligned} \frac{d F}{d u^n} & = & (1/\delta t)F_{\dot{u}} + F_u(t^n,u^n,\dot{u}^n).\end{aligned}\]

    But \(F_{\dot{u}} = 1\), in this special case, resulting in the expected Jacobian \(I/\delta t - f_u(t,u^n)\).

  • Jacobian \(G_u\)
    If using a fully implicit method and the function \(G()\) is provided, one also can provide an appropriate (approximate) Jacobian matrix of \(G()\).
    TSSetRHSJacobian(TS ts,Mat A,Mat B,
PetscErrorCode (*fjac)(TS,PetscReal,Vec,Mat,Mat,void*),void *jacP);

    The arguments for the function fjac() are the timestep context, current time, input state \(u\), matrix \(A\), preconditioning matrix \(B\), and the (optional) user-provided context jacP.

Providing appropriate \(F()\) and \(G()\) for your problem allows for the easy runtime switching between explicit, semi-implicit (IMEX), and fully implicit methods.

Basic TS Options#

The user first creates a TS object with the command

int TSCreate(MPI_Comm comm,TS *ts);

int TSSetProblemType(TS ts,TSProblemType problemtype);

The TSProblemType is one of TS_LINEAR or TS_NONLINEAR.

To set up TS for solving an ODE, one must set the “initial conditions” for the ODE with

TSSetSolution(TS ts, Vec initialsolution);

One can set the solution method with the routine

TSSetType(TS ts,TSType type);
Currently supported types are TSEULER, TSRK (Runge-Kutta), TSBEULER, TSCN (Crank-Nicolson), TSTHETA, TSGLLE (generalized linear), TSPSEUDO, and TSSUNDIALS (only if the Sundials package is installed), or the command line option
-ts_type euler,rk,beuler,cn,theta,gl,pseudo,sundials,eimex,arkimex,rosw.

A list of available methods is given in Summary of Time Integrators Available In PETSc.

Set the initial time with the command

TSSetTime(TS ts,PetscReal time);

One can change the timestep with the command

TSSetTimeStep(TS ts,PetscReal dt);

can determine the current timestep with the routine

TSGetTimeStep(TS ts,PetscReal* dt);

Here, “current” refers to the timestep being used to attempt to promote the solution form \(u^n\) to \(u^{n+1}.\)

One sets the total number of timesteps to run or the total time to run (whatever is first) with the commands

TSSetMaxSteps(TS ts,PetscInt maxsteps);
TSSetMaxTime(TS ts,PetscReal maxtime);

and determines the behavior near the final time with

TSSetExactFinalTime(TS ts,TSExactFinalTimeOption eftopt);

where eftopt is one of TS_EXACTFINALTIME_STEPOVER,TS_EXACTFINALTIME_INTERPOLATE, or TS_EXACTFINALTIME_MATCHSTEP. One performs the requested number of time steps with

TSSolve(TS ts,Vec U);

The solve call implicitly sets up the timestep context; this can be done explicitly with

TSSetUp(TS ts);

One destroys the context with

TSDestroy(TS *ts);

and views it with

TSView(TS ts,PetscViewer viewer);

In place of TSSolve(), a single step can be taken using

TSStep(TS ts);

DAE Formulations#

You can find a discussion of DAEs in [AP98] or Scholarpedia. In PETSc, TS deals with the semi-discrete form of the equations, so that space has already been discretized. If the DAE depends explicitly on the coordinate \(x\), then this will just appear as any other data for the equation, not as an explicit argument. Thus we have

\[F(t, u, \dot{u}) = 0\]

In this form, only fully implicit solvers are appropriate. However, specialized solvers for restricted forms of DAE are supported by PETSc. Below we consider an ODE which is augmented with algebraic constraints on the variables.

Hessenberg Index-1 DAE#

This is a Semi-Explicit Index-1 DAE which has the form

\[\begin{aligned} \dot{u} &= f(t, u, z) \\ 0 &= h(t, u, z) \end{aligned}\]

where \(z\) is a new constraint variable, and the Jacobian \(\frac{dh}{dz}\) is non-singular everywhere. We have suppressed the \(x\) dependence since it plays no role here. Using the non-singularity of the Jacobian and the Implicit Function Theorem, we can solve for \(z\) in terms of \(u\). This means we could, in principle, plug \(z(u)\) into the first equation to obtain a simple ODE, even if this is not the numerical process we use. Below we show that this type of DAE can be used with IMEX schemes.

Hessenberg Index-2 DAE#

This DAE has the form

\[\begin{aligned} \dot{u} &= f(t, u, z) \\ 0 &= h(t, u) \end{aligned}\]

Notice that the constraint equation \(h\) is not a function of the constraint variable \(z\). This means that we cannot naively invert as we did in the index-1 case. Our strategy will be to convert this into an index-1 DAE using a time derivative, which loosely corresponds to the idea of an index being the number of derivatives necessary to get back to an ODE. If we differentiate the constraint equation with respect to time, we can use the ODE to simplify it,

\[\begin{aligned} 0 &= \dot{h}(t, u) \\ &= \frac{dh}{du} \dot{u} + \frac{\partial h}{\partial t} \\ &= \frac{dh}{du} f(t, u, z) + \frac{\partial h}{\partial t} \end{aligned}\]

If the Jacobian \(\frac{dh}{du} \frac{df}{dz}\) is non-singular, then we have precisely a semi-explicit index-1 DAE, and we can once again use the PETSc IMEX tools to solve it. A common example of an index-2 DAE is the incompressible Navier-Stokes equations, since the continuity equation \(\nabla\cdot u = 0\) does not involve the pressure. Using PETSc IMEX with the above conversion then corresponds to the Segregated Runge-Kutta method applied to this equation [ColomesB16].

Using Implicit-Explicit (IMEX) Methods#

For “stiff” problems or those with multiple time scales \(F()\) will be treated implicitly using a method suitable for stiff problems and \(G()\) will be treated explicitly when using an IMEX method like TSARKIMEX. \(F()\) is typically linear or weakly nonlinear while \(G()\) may have very strong nonlinearities such as arise in non-oscillatory methods for hyperbolic PDE. The user provides three pieces of information, the APIs for which have been described above.

The user needs to set TSSetEquationType() to TS_EQ_IMPLICIT or higher if the problem is implicit; e.g., \(F(t,u,\dot u) = M \dot u - f(t,u)\), where \(M\) is not the identity matrix:

  • the problem is an implicit ODE (defined implicitly through TSSetIFunction()) or

  • a DAE is being solved.

An IMEX problem representation can be made implicit by setting TSARKIMEXSetFullyImplicit().

In PETSc, DAEs and ODEs are formulated as \(F(t,u,\dot{u})=G(t,u)\), where \(F()\) is meant to be integrated implicitly and \(G()\) explicitly. An IMEX formulation such as \(M\dot{u}=f(t,u)+g(t,u)\) requires the user to provide \(M^{-1} g(t,u)\) or solve \(g(t,u) - M x=0\) in place of \(G(t,u)\). General cases such as \(F(t,u,\dot{u})=G(t,u)\) are not amenable to IMEX Runge-Kutta, but can be solved by using fully implicit methods. Some use-case examples for TSARKIMEX are listed in Table 13 and a list of methods with a summary of their properties is given in IMEX Runge-Kutta schemes.

Table 13 Use case examples for TSARKIMEX#

\(\dot{u} = g(t,u)\)

nonstiff ODE

\(\begin{aligned}F(t,u,\dot{u}) &= \dot{u} \\ G(t,u) &= g(t,u)\end{aligned}\)

\(M \dot{u} = g(t,u)\)

nonstiff ODE with mass matrix

\(\begin{aligned}F(t,u,\dot{u}) &= \dot{u} \\ G(t,u) &= M^{-1} g(t,u)\end{aligned}\)

\(\dot{u} = f(t,u)\)

stiff ODE

\(\begin{aligned}F(t,u,\dot{u}) &= \dot{u} - f(t,u) \\ G(t,u) &= 0\end{aligned}\)

\(M \dot{u} = f(t,u)\)

stiff ODE with mass matrix

\(\begin{aligned}F(t,u,\dot{u}) &= M \dot{u} - f(t,u) \\ G(t,u) &= 0\end{aligned}\)

\(\dot{u} = f(t,u) + g(t,u)\)

stiff-nonstiff ODE

\(\begin{aligned}F(t,u,\dot{u}) &= \dot{u} - f(t,u) \\ G(t,u) &= g(t,u)\end{aligned}\)

\(M \dot{u} = f(t,u) + g(t,u)\)

stiff-nonstiff ODE with mass matrix

\(\begin{aligned}F(t,u,\dot{u}) &= M\dot{u} - f(t,u) \\ G(t,u) &= M^{-1} g(t,u)\end{aligned}\)

\(\begin{aligned}\dot{u} &= f(t,u,z) + g(t,u,z)\\0 &= h(t,y,z)\end{aligned}\)

semi-explicit index-1 DAE

\(\begin{aligned}F(t,u,\dot{u}) &= \begin{pmatrix}\dot{u} - f(t,u,z)\\h(t, u, z)\end{pmatrix}\\G(t,u) &= g(t,u)\end{aligned}\)

\(f(t,u,\dot{u})=0\)

fully implicit ODE/DAE

\(\begin{aligned}F(t,u,\dot{u}) &= f(t,u,\dot{u})\\G(t,u) &= 0\end{aligned}\); the user needs to set TSSetEquationType() to TS_EQ_IMPLICIT or higher

Table 14 lists of the currently available IMEX Runge-Kutta schemes. For each method, it gives the -ts_arkimex_type name, the reference, the total number of stages/implicit stages, the order/stage-order, the implicit stability properties (IM), stiff accuracy (SA), the existence of an embedded scheme, and dense output (DO).

Table 14 IMEX Runge-Kutta schemes#

Name

Reference

Stages (IM)

Order (Stage)

IM

SA

Embed

DO

Remarks

a2

based on CN

2 (1)

2 (2)

A-Stable

yes

yes (1)

yes (2)

l2

SSP2(2,2,2) [PR05]

2 (2)

2 (1)

L-Stable

yes

yes (1)

yes (2)

SSP SDIRK

ars122

ARS122 [ARS97]

2 (1)

3 (1)

A-Stable

yes

yes (1)

yes (2)

2c

[GKC13]

3 (2)

2 (2)

L-Stable

yes

yes (1)

yes (2)

SDIRK

2d

[GKC13]

3 (2)

2 (2)

L-Stable

yes

yes (1)

yes (2)

SDIRK

2e

[GKC13]

3 (2)

2 (2)

L-Stable

yes

yes (1)

yes (2)

SDIRK

prssp2

PRS(3,3,2) [PR05]

3 (3)

3 (1)

L-Stable

yes

no

no

SSP

3

[KC03]

4 (3)

3 (2)

L-Stable

yes

yes (2)

yes (2)

SDIRK

bpr3

[BPR11]

5 (4)

3 (2)

L-Stable

yes

no

no

SDIRK

ars443

[ARS97]

5 (4)

3 (1)

L-Stable

yes

no

no

SDIRK

4

[KC03]

6 (5)

4 (2)

L-Stable

yes

yes (3)

yes

SDIRK

5

[KC03]

8 (7)

5 (2)

L-Stable

yes

yes (4)

yes (3)

SDIRK

ROSW are linearized implicit Runge-Kutta methods known as Rosenbrock W-methods. They can accommodate inexact Jacobian matrices in their formulation. A series of methods are available in PETSc are listed in Table 15 below. For each method, it gives the reference, the total number of stages and implicit stages, the scheme order and stage order, the implicit stability properties (IM), stiff accuracy (SA), the existence of an embedded scheme, dense output (DO), the capacity to use inexact Jacobian matrices (-W), and high order integration of differential algebraic equations (PDAE).

Table 15 Rosenbrock W-schemes#

TS

Reference

Stages (IM)

Order (Stage)

IM

SA

Embed

DO

-W

PDAE

Remarks

theta1

classical

1(1)

1(1)

L-Stable

theta2

classical

1(1)

2(2)

A-Stable

2m

Zoltan

2(2)

2(1)

L-Stable

No

Yes(1)

Yes(2)

Yes

No

SSP

2p

Zoltan

2(2)

2(1)

L-Stable

No

Yes(1)

Yes(2)

Yes

No

SSP

ra3pw

[RA05]

3(3)

3(1)

A-Stable

No

Yes

Yes(2)

No

Yes(3)

ra34pw2

[RA05]

4(4)

3(1)

L-Stable

Yes

Yes

Yes(3)

Yes

Yes(3)

rodas3

[SVB+97]

4(4)

3(1)

L-Stable

Yes

Yes

No

No

Yes

sandu3

[SVB+97]

3(3)

3(1)

L-Stable

Yes

Yes

Yes(2)

No

No

assp3p3s1c

unpub.

3(2)

3(1)

A-Stable

No

Yes

Yes(2)

Yes

No

SSP

lassp3p4s2c

unpub.

4(3)

3(1)

L-Stable

No

Yes

Yes(3)

Yes

No

SSP

lassp3p4s2c

unpub.

4(3)

3(1)

L-Stable

No

Yes

Yes(3)

Yes

No

SSP

ark3

unpub.

4(3)

3(1)

L-Stable

No

Yes

Yes(3)

Yes

No

IMEX-RK

GLEE methods#

In this section, we describe explicit and implicit time stepping methods with global error estimation that are introduced in [Con16]. The solution vector for a GLEE method is either [\(y\), \(\tilde{y}\)] or [\(y\),\(\varepsilon\)], where \(y\) is the solution, \(\tilde{y}\) is the “auxiliary solution,” and \(\varepsilon\) is the error. The working vector that TSGLEE uses is \(Y\) = [\(y\),\(\tilde{y}\)], or [\(y\),\(\varepsilon\)]. A GLEE method is defined by

  • \((p,r,s)\): (order, steps, and stages),

  • \(\gamma\): factor representing the global error ratio,

  • \(A, U, B, V\): method coefficients,

  • \(S\): starting method to compute the working vector from the solution (say at the beginning of time integration) so that \(Y = Sy\),

  • \(F\): finalizing method to compute the solution from the working vector,\(y = FY\).

  • \(F_\text{embed}\): coefficients for computing the auxiliary solution \(\tilde{y}\) from the working vector (\(\tilde{y} = F_\text{embed} Y\)),

  • \(F_\text{error}\): coefficients to compute the estimated error vector from the working vector (\(\varepsilon = F_\text{error} Y\)).

  • \(S_\text{error}\): coefficients to initialize the auxiliary solution (\(\tilde{y}\) or \(\varepsilon\)) from a specified error vector (\(\varepsilon\)). It is currently implemented only for \(r = 2\). We have \(y_\text{aux} = S_{error}[0]*\varepsilon + S_\text{error}[1]*y\), where \(y_\text{aux}\) is the 2nd component of the working vector \(Y\).

The methods can be described in two mathematically equivalent forms: propagate two components (“\(y\tilde{y}\) form”) and propagating the solution and its estimated error (“\(y\varepsilon\) form”). The two forms are not explicitly specified in TSGLEE; rather, the specific values of \(B, U, S, F, F_{embed}\), and \(F_{error}\) characterize whether the method is in \(y\tilde{y}\) or \(y\varepsilon\) form.

The API used by this TS method includes:

  • TSGetSolutionComponents: Get all the solution components of the working vector

    ierr = TSGetSolutionComponents(TS,int*,Vec*)

    Call with NULL as the last argument to get the total number of components in the working vector \(Y\) (this is \(r\) (not \(r-1\))), then call to get the \(i\)-th solution component.

  • TSGetAuxSolution: Returns the auxiliary solution \(\tilde{y}\) (computed as \(F_\text{embed} Y\))

    ierr = TSGetAuxSolution(TS,Vec*)

  • TSGetTimeError: Returns the estimated error vector \(\varepsilon\) (computed as \(F_\text{error} Y\) if \(n=0\) or restores the error estimate at the end of the previous step if \(n=-1\))

    ierr = TSGetTimeError(TS,PetscInt n,Vec*)

  • TSSetTimeError: Initializes the auxiliary solution (\(\tilde{y}\) or \(\varepsilon\)) for a specified initial error.

    ierr = TSSetTimeError(TS,Vec)

The local error is estimated as \(\varepsilon(n+1)-\varepsilon(n)\). This is to be used in the error control. The error in \(y\tilde{y}\) GLEE is \(\varepsilon(n) = \frac{1}{1-\gamma} * (\tilde{y}(n) - y(n))\).

Note that \(y\) and \(\tilde{y}\) are reported to TSAdapt basic (TSADAPTBASIC), and thus it computes the local error as \(\varepsilon_{loc} = (\tilde{y} - y)\). However, the actual local error is \(\varepsilon_{loc} = \varepsilon_{n+1} - \varepsilon_n = \frac{1}{1-\gamma} * [(\tilde{y} - y)_{n+1} - (\tilde{y} - y)_n]\).

Table 16 lists currently available GL schemes with global error estimation [Con16].

Table 16 GL schemes with global error estimation#

TS

Reference

IM/EX

\((p,r,s)\)

\(\gamma\)

Form

Notes

TSGLEEi1

BE1

IM

\((1,3,2)\)

\(0.5\)

\(y\varepsilon\)

Based on backward Euler

TSGLEE23

23

EX

\((2,3,2)\)

\(0\)

\(y\varepsilon\)

TSGLEE24

24

EX

\((2,4,2)\)

\(0\)

\(y\tilde{y}\)

TSGLEE25I

25i

EX

\((2,5,2)\)

\(0\)

\(y\tilde{y}\)

TSGLEE35

35

EX

\((3,5,2)\)

\(0\)

\(y\tilde{y}\)

TSGLEEEXRK2A

exrk2a

EX

\((2,6,2)\)

\(0.25\)

\(y\varepsilon\)

TSGLEERK32G1

rk32g1

EX

\((3,8,2)\)

\(0\)

\(y\varepsilon\)

TSGLEERK285EX

rk285ex

EX

\((2,9,2)\)

\(0.25\)

\(y\varepsilon\)

Using fully implicit methods#

To use a fully implicit method like TSTHETA or TSGL, either provide the Jacobian of \(F()\) (and \(G()\) if \(G()\) is provided) or use a DM that provides a coloring so the Jacobian can be computed efficiently via finite differences.

Using the Explicit Runge-Kutta timestepper with variable timesteps#

The explicit Euler and Runge-Kutta methods require the ODE be in the form

\[\dot{u} = G(u,t). \]

The user can either call TSSetRHSFunction() and/or they can call TSSetIFunction() (so long as the function provided to TSSetIFunction() is equivalent to \(\dot{u} + \tilde{F}(t,u)\)) but the Jacobians need not be provided. 5

The Explicit Runge-Kutta timestepper with variable timesteps is an implementation of the standard Runge-Kutta with an embedded method. The error in each timestep is calculated using the solutions from the Runge-Kutta method and its embedded method (the 2-norm of the difference is used). The default method is the \(3\)rd-order Bogacki-Shampine method with a \(2\)nd-order embedded method (TSRK3BS). Other available methods are the \(5\)th-order Fehlberg RK scheme with a \(4\)th-order embedded method (TSRK5F), the \(5\)th-order Dormand-Prince RK scheme with a \(4\)th-order embedded method (TSRK5DP), the \(5\)th-order Bogacki-Shampine RK scheme with a \(4\)th-order embedded method (TSRK5BS, and the \(6\)th-, \(7\)th, and \(8\)th-order robust Verner RK schemes with a \(5\)th-, \(6\)th, and \(7\)th-order embedded method, respectively (TSRK6VR, TSRK7VR, TSRK8VR). Variable timesteps cannot be used with RK schemes that do not have an embedded method (TSRK1FE - \(1\)st-order, \(1\)-stage forward Euler, TSRK2A - \(2\)nd-order, \(2\)-stage RK scheme, TSRK3 - \(3\)rd-order, \(3\)-stage RK scheme, TSRK4 - \(4\)-th order, \(4\)-stage RK scheme).

Special Cases#

Monitoring and visualizing solutions#

  • -ts_monitor - prints the time and timestep at each iteration.

  • -ts_adapt_monitor - prints information about the timestep adaption calculation at each iteration.

  • -ts_monitor_lg_timestep - plots the size of each timestep, TSMonitorLGTimeStep().

  • -ts_monitor_lg_solution - for ODEs with only a few components (not arising from the discretization of a PDE) plots the solution as a function of time, TSMonitorLGSolution().

  • -ts_monitor_lg_error - for ODEs with only a few components plots the error as a function of time, only if TSSetSolutionFunction() is provided, TSMonitorLGError().

  • -ts_monitor_draw_solution - plots the solution at each iteration, TSMonitorDrawSolution().

  • -ts_monitor_draw_error - plots the error at each iteration only if TSSetSolutionFunction() is provided, TSMonitorDrawSolution().

  • -ts_monitor_solution binary[:filename] - saves the solution at each iteration to a binary file, TSMonitorSolution(). Solution viewers work with other time-aware formats, e.g., -ts_monitor_solution cgns:sol.cgns, and can output one solution every 10 time steps by adding -ts_monitor_solution_interval 10.

  • -ts_monitor_solution_vtk <filename-%03D.vts> - saves the solution at each iteration to a file in vtk format, TSMonitorSolutionVTK().

Error control via variable time-stepping#

Most of the time stepping methods avaialable in PETSc have an error estimation and error control mechanism. This mechanism is implemented by changing the step size in order to maintain user specified absolute and relative tolerances. The PETSc object responsible with error control is TSAdapt. The available TSAdapt types are listed in the following table.

Table 17 TSAdapt: available adaptors#

ID

Name

Notes

TSADAPTNONE

none

no adaptivity

TSADAPTBASIC

basic

the default adaptor

TSADAPTGLEE

glee

extension of the basic adaptor to treat \({\rm Tol}_{\rm A}\) and \({\rm Tol}_{\rm R}\) as separate criteria. It can also control global erorrs if the integrator (e.g., TSGLEE) provides this information

When using TSADAPTBASIC (the default), the user typically provides a desired absolute \({\rm Tol}_{\rm A}\) or a relative \({\rm Tol}_{\rm R}\) error tolerance by invoking TSSetTolerances() or at the command line with options -ts_atol and -ts_rtol. The error estimate is based on the local truncation error, so for every step the algorithm verifies that the estimated local truncation error satisfies the tolerances provided by the user and computes a new step size to be taken. For multistage methods, the local truncation is obtained by comparing the solution \(y\) to a lower order \(\widehat{p}=p-1\) approximation, \(\widehat{y}\), where \(p\) is the order of the method and \(\widehat{p}\) the order of \(\widehat{y}\).

The adaptive controller at step \(n\) computes a tolerance level

\[\begin{aligned} Tol_n(i)&=&{\rm Tol}_{\rm A}(i) + \max(y_n(i),\widehat{y}_n(i)) {\rm Tol}_{\rm R}(i)\,,\end{aligned}\]

and forms the acceptable error level

\[\begin{aligned} \rm wlte_n&=& \frac{1}{m} \sum_{i=1}^{m}\sqrt{\frac{\left\|y_n(i) -\widehat{y}_n(i)\right\|}{Tol(i)}}\,,\end{aligned}\]

where the errors are computed componentwise, \(m\) is the dimension of \(y\) and -ts_adapt_wnormtype is 2 (default). If -ts_adapt_wnormtype is infinity (max norm), then

\[\begin{aligned} \rm wlte_n&=& \max_{1\dots m}\frac{\left\|y_n(i) -\widehat{y}_n(i)\right\|}{Tol(i)}\,.\end{aligned}\]

The error tolerances are satisfied when \(\rm wlte\le 1.0\).

The next step size is based on this error estimate, and determined by

(5)#\[\begin{aligned} \Delta t_{\rm new}(t)&=&\Delta t_{\rm{old}} \min(\alpha_{\max}, \max(\alpha_{\min}, \beta (1/\rm wlte)^\frac{1}{\widehat{p}+1}))\,,\end{aligned}\]

where \(\alpha_{\min}=\)-ts_adapt_clip[0] and \(\alpha_{\max}\)=-ts_adapt_clip[1] keep the change in \(\Delta t\) to within a certain factor, and \(\beta<1\) is chosen through -ts_adapt_safety so that there is some margin to which the tolerances are satisfied and so that the probability of rejection is decreased.

This adaptive controller works in the following way. After completing step \(k\), if \(\rm wlte_{k+1} \le 1.0\), then the step is accepted and the next step is modified according to eq:hnew; otherwise, the step is rejected and retaken with the step length computed in (5).

TSADAPTGLEE is an extension of the basic adaptor to treat \({\rm Tol}_{\rm A}\) and \({\rm Tol}_{\rm R}\) as separate criteria. it can also control global errors if the integrator (e.g., TSGLEE) provides this information.

Handling of discontinuities#

For problems that involve discontinuous right hand sides, one can set an “event” function \(g(t,u)\) for PETSc to detect and locate the times of discontinuities (zeros of \(g(t,u)\)). Events can be defined through the event monitoring routine

TSSetEventHandler(TS ts,PetscInt nevents,PetscInt *direction,PetscBool *terminate,PetscErrorCode (*eventhandler)(TS,PetscReal,Vec,PetscScalar*,void* eventP),PetscErrorCode (*postevent)(TS,PetscInt,PetscInt[],PetscReal,Vec,PetscBool,void* eventP),void *eventP);

Here, nevents denotes the number of events, direction sets the type of zero crossing to be detected for an event (+1 for positive zero-crossing, -1 for negative zero-crossing, and 0 for both), terminate conveys whether the time-stepping should continue or halt when an event is located, eventmonitor is a user- defined routine that specifies the event description, postevent is an optional user-defined routine to take specific actions following an event.

The arguments to eventhandler() are the timestep context, current time, input state \(u\), array of event function value, and the (optional) user-provided context eventP.

The arguments to postevent() routine are the timestep context, number of events occurred, indices of events occured, current time, input state \(u\), a boolean flag indicating forward solve (1) or adjoint solve (0), and the (optional) user-provided context eventP.

The event monitoring functionality is only available with PETSc’s implicit time-stepping solvers TSTHETA, TSARKIMEX, and TSROSW.

Explicit integrators with finite element mass matrices#

Discretized finite element problems often have the form \(M \dot u = G(t, u)\) where \(M\) is the mass matrix. Such problems can be solved using DMTSSetIFunction() with implicit integrators. When \(M\) is nonsingular (i.e., the problem is an ODE, not a DAE), explicit integrators can be applied to \(\dot u = M^{-1} G(t, u)\) or \(\dot u = \hat M^{-1} G(t, u)\), where \(\hat M\) is the lumped mass matrix. While the true mass matrix generally has a dense inverse and thus must be solved iteratively, the lumped mass matrix is diagonal (e.g., computed via collocated quadrature or row sums of \(M\)). To have PETSc create and apply a (lumped) mass matrix automatically, first use DMTSSetRHSFunction()` to specify :math:`G` and set a ``PetscFE` using ``DMAddField() and DMCreateDS(), then call either DMTSCreateRHSMassMatrix() or DMTSCreateRHSMassMatrixLumped() to automatically create the mass matrix and a KSP that will be used to apply \(M^{-1}\). This KSP can be customized using the "mass_" prefix.

Performing sensitivity analysis with the TS ODE Solvers#

The TS library provides a framework based on discrete adjoint models for sensitivity analysis for ODEs and DAEs. The ODE/DAE solution process (henceforth called the forward run) can be obtained by using either explicit or implicit solvers in TS, depending on the problem properties. Currently supported method types are TSRK (Runge-Kutta) explicit methods and TSTHETA implicit methods, which include TSBEULER and TSCN.

Using the discrete adjoint methods#

Consider the ODE/DAE

\[F(t,y,\dot{y},p) = 0, \quad y(t_0)=y_0(p) \quad t_0 \le t \le t_F \]

and the cost function(s)

\[\Psi_i(y_0,p) = \Phi_i(y_F,p) + \int_{t_0}^{t_F} r_i(y(t),p,t)dt \quad i=1,...,n_\text{cost}. \]

The TSAdjoint routines of PETSc provide

\[\frac{\partial \Psi_i}{\partial y_0} = \lambda_i \]

and

\[\frac{\partial \Psi_i}{\partial p} = \mu_i + \lambda_i (\frac{\partial y_0}{\partial p}). \]

To perform the discrete adjoint sensitivity analysis one first sets up the TS object for a regular forward run but with one extra function call

TSSetSaveTrajectory(TS ts),

then calls TSSolve() in the usual manner.

One must create two arrays of \(n_\text{cost}\) vectors \(\lambda\) and \(\mu\) (if there are no parameters \(p\) then one can use NULL for the \(\mu\) array.) The \(\lambda\) vectors are the same dimension and parallel layout as the solution vector for the ODE, the \(\mu\) vectors are of dimension \(p\); when \(p\) is small usually all its elements are on the first MPI process, while the vectors have no entries on the other processes. \(\lambda_i\) and \(\mu_i\) should be initialized with the values \(d\Phi_i/dy|_{t=t_F}\) and \(d\Phi_i/dp|_{t=t_F}\) respectively. Then one calls

TSSetCostGradients(TS ts,PetscInt numcost, Vec *lambda,Vec *mu);

where numcost denotes \(n_\text{cost}\). If \(F()\) is a function of \(p\) one needs to also provide the Jacobian \(-F_p\) with

TSSetRHSJacobianP(TS ts,Mat Amat,PetscErrorCode (*fp)(TS,PetscReal,Vec,Mat,void*),void *ctx)

or

TSSetIJacobianP(TS ts,Mat Amat,PetscErrorCode (*fp)(TS,PetscReal,Vec,Vec,PetscReal,Mat,void*),void *ctx)

or both, depending on which form is used to define the ODE.

The arguments for the function fp() are the timestep context, current time, \(y\), and the (optional) user-provided context.

If there is an integral term in the cost function, i.e. \(r\) is nonzero, it can be transformed into another ODE that is augmented to the original ODE. To evaluate the integral, one needs to create a child TS objective by calling

TSCreateQuadratureTS(TS ts,PetscBool fwd,TS *quadts);

and provide the ODE RHS function (which evaluates the integrand \(r\)) with

TSSetRHSFunction(TS quadts,Vec R,PetscErrorCode (*rf)(TS,PetscReal,Vec,Vec,void*),void *ctx)

Similar to the settings for the original ODE, Jacobians of the integrand can be provided with

TSSetRHSJacobian(TS quadts,Vec DRDU,Vec DRDU,PetscErrorCode (*drdyf)(TS,PetscReal,Vec,Vec*,void*),void *ctx)
TSSetRHSJacobianP(TS quadts,Vec DRDU,Vec DRDU,PetscErrorCode (*drdyp)(TS,PetscReal,Vec,Vec*,void*),void *ctx)

where \(\mathrm{drdyf}= dr /dy\), \(\mathrm{drdpf} = dr /dp\). Since the integral term is additive to the cost function, its gradient information will be included in \(\lambda\) and \(\mu\).

Lastly, one starts the backward run by calling

TSAdjointSolve(TS ts).

One can obtain the value of the integral term by calling

TSGetCostIntegral(TS ts,Vec *q).

or accessing directly the solution vector used by quadts.

The second argument of TSCreateQuadratureTS() allows one to choose if the integral term is evaluated in the forward run (inside TSSolve()) or in the backward run (inside TSAdjointSolve()) when TSSetCostGradients() and TSSetCostIntegrand() are called before TSSolve(). Note that this also allows for evaluating the integral without having to use the adjoint solvers.

To provide a better understanding of the use of the adjoint solvers, we introduce a simple example, corresponding to TS Power Grid Tutorial ex3sa. The problem is to study dynamic security of power system when there are credible contingencies such as short-circuits or loss of generators, transmission lines, or loads. The dynamic security constraints are incorporated as equality constraints in the form of discretized differential equations and inequality constraints for bounds on the trajectory. The governing ODE system is

\[\begin{aligned} \phi' &= &\omega_B (\omega - \omega_S) \\ 2H/\omega_S \, \omega' & =& p_m - p_{max} sin(\phi) -D (\omega - \omega_S), \quad t_0 \leq t \leq t_F,\end{aligned}\]

where \(\phi\) is the phase angle and \(\omega\) is the frequency.

The initial conditions at time \(t_0\) are

\[\begin{aligned} \phi(t_0) &=& \arcsin \left( p_m / p_{max} \right), \\ w(t_0) & =& 1.\end{aligned}\]

\(p_{max}\) is a positive number when the system operates normally. At an event such as fault incidence/removal, \(p_{max}\) will change to \(0\) temporarily and back to the original value after the fault is fixed. The objective is to maximize \(p_m\) subject to the above ODE constraints and \(\phi<\phi_S\) during all times. To accommodate the inequality constraint, we want to compute the sensitivity of the cost function

\[\Psi(p_m,\phi) = -p_m + c \int_{t_0}^{t_F} \left( \max(0, \phi - \phi_S ) \right)^2 dt \]

with respect to the parameter \(p_m\). \(numcost\) is \(1\) since it is a scalar function.

For ODE solution, PETSc requires user-provided functions to evaluate the system \(F(t,y,\dot{y},p)\) (set by TSSetIFunction() ) and its corresponding Jacobian \(F_y + \sigma F_{\dot y}\) (set by TSSetIJacobian()). Note that the solution state \(y\) is \([ \phi \; \omega ]^T\) here. For sensitivity analysis, we need to provide a routine to compute \(\mathrm{f}_p=[0 \; 1]^T\) using TSASetRHSJacobianP(), and three routines corresponding to the integrand \(r=c \left( \max(0, \phi - \phi_S ) \right)^2\), \(r_p = [0 \; 0]^T\) and \(r_y= [ 2 c \left( \max(0, \phi - \phi_S ) \right) \; 0]^T\) using TSSetCostIntegrand().

In the adjoint run, \(\lambda\) and \(\mu\) are initialized as \([ 0 \; 0 ]^T\) and \([-1]\) at the final time \(t_F\). After TSAdjointSolve(), the sensitivity of the cost function w.r.t. initial conditions is given by the sensitivity variable \(\lambda\) (at time \(t_0\)) directly. And the sensitivity of the cost function w.r.t. the parameter \(p_m\) can be computed (by users) as

\[\frac{\mathrm{d} \Psi}{\mathrm{d} p_m} = \mu(t_0) + \lambda(t_0) \frac{\mathrm{d} \left[ \phi(t_0) \; \omega(t_0) \right]^T}{\mathrm{d} p_m} . \]

For explicit methods where one does not need to provide the Jacobian \(F_u\) for the forward solve one still does need it for the backward solve and thus must call

TSSetRHSJacobian(TS ts,Mat Amat, Mat Pmat,PetscErrorCode (*f)(TS,PetscReal,Vec,Mat,Mat,void*),void *fP);

Examples include:

Checkpointing#

The discrete adjoint model requires the states (and stage values in the context of multistage timestepping methods) to evaluate the Jacobian matrices during the adjoint (backward) run. By default, PETSc stores the whole trajectory to disk as binary files, each of which contains the information for a single time step including state, time, and stage values (optional). One can also make PETSc store the trajectory to memory with the option -ts_trajectory_type memory. However, there might not be sufficient memory capacity especially for large-scale problems and long-time integration.

A so-called checkpointing scheme is needed to solve this problem. The scheme stores checkpoints at selective time steps and recomputes the missing information. The revolve library is used by PETSc TSTrajectory to generate an optimal checkpointing schedule that minimizes the recomputations given a limited number of available checkpoints. One can specify the number of available checkpoints with the option -ts_trajectory_max_cps_ram [maximum number of checkpoints in RAM]. Note that one checkpoint corresponds to one time step.

The revolve library also provides an optimal multistage checkpointing scheme that uses both RAM and disk for storage. This scheme is automatically chosen if one uses both the option -ts_trajectory_max_cps_ram [maximum number of checkpoints in RAM] and the option -ts_trajectory_max_cps_disk [maximum number of checkpoints on disk].

Some other useful options are listed below.

  • -ts_trajectory_view prints the total number of recomputations,

  • -ts_monitor and -ts_adjoint_monitor allow users to monitor the progress of the adjoint work flow,

  • -ts_trajectory_type visualization may be used to save the whole trajectory for visualization. It stores the solution and the time, but no stage values. The binary files generated can be read into MATLAB via the script $PETSC_DIR/share/petsc/matlab/PetscReadBinaryTrajectory.m.

Using Sundials from PETSc#

Sundials is a parallel ODE solver developed by Hindmarsh et al. at LLNL. The TS library provides an interface to use the CVODE component of Sundials directly from PETSc. (To configure PETSc to use Sundials, see the installation guide, installation/index.htm.)

To use the Sundials integrators, call

TSSetType(TS ts,TSType TSSUNDIALS);

or use the command line option -ts_type sundials.

Sundials’ CVODE solver comes with two main integrator families, Adams and BDF (backward differentiation formula). One can select these with

TSSundialsSetType(TS ts,TSSundialsLmmType [SUNDIALS_ADAMS,SUNDIALS_BDF]);

or the command line option -ts_sundials_type <adams,bdf>. BDF is the default.

Sundials does not use the SNES library within PETSc for its nonlinear solvers, so one cannot change the nonlinear solver options via SNES. Rather, Sundials uses the preconditioners within the PC package of PETSc, which can be accessed via

TSSundialsGetPC(TS ts,PC *pc);

The user can then directly set preconditioner options; alternatively, the usual runtime options can be employed via -pc_xxx.

Finally, one can set the Sundials tolerances via

TSSundialsSetTolerance(TS ts,double abs,double rel);

where abs denotes the absolute tolerance and rel the relative tolerance.

Other PETSc-Sundials options include

TSSundialsSetGramSchmidtType(TS ts,TSSundialsGramSchmidtType type);

where type is either SUNDIALS_MODIFIED_GS or SUNDIALS_UNMODIFIED_GS. This may be set via the options data base with -ts_sundials_gramschmidt_type <modifed,unmodified>.

The routine

TSSundialsSetMaxl(TS ts,PetscInt restart);

sets the number of vectors in the Krylov subpspace used by GMRES. This may be set in the options database with -ts_sundials_maxl maxl.

Using TChem from PETSc#

TChem 6 is a package originally developed at Sandia National Laboratory that can read in CHEMKIN 7 data files and compute the right hand side function and its Jacobian for a reaction ODE system. To utilize PETSc’s ODE solvers for these systems, first install PETSc with the additional configure option --download-tchem. We currently provide two examples of its use; one for single cell reaction and one for an “artificial” one dimensional problem with periodic boundary conditions and diffusion of all species. The self-explanatory examples are the The TS tutorial extchem and The TS tutorial extchemfield.

4

If the matrix \(F_{\dot{u}}(t) = \partial F / \partial \dot{u}\) is nonsingular then it is an ODE and can be transformed to the standard explicit form, although this transformation may not lead to efficient algorithms.

5

PETSc will automatically translate the function provided to the appropriate form.

6

bitbucket.org/jedbrown/tchem

7

en.wikipedia.org/wiki/CHEMKIN

Solving Steady-State Problems with Pseudo-Timestepping#

Simple Example: TS provides a general code for performing pseudo timestepping with a variable timestep at each physical node point. For example, instead of directly attacking the steady-state problem

\[G(u) = 0, \]

we can use pseudo-transient continuation by solving

\[u_t = G(u). \]

Using time differencing

\[u_t \doteq \frac{{u^{n+1}} - {u^{n}} }{dt^{n}} \]

with the backward Euler method, we obtain nonlinear equations at a series of pseudo-timesteps

\[\frac{1}{dt^n} B (u^{n+1} - u^{n} ) = G(u^{n+1}). \]

For this problem the user must provide \(G(u)\), the time steps \(dt^{n}\) and the left-hand-side matrix \(B\) (or optionally, if the timestep is position independent and \(B\) is the identity matrix, a scalar timestep), as well as optionally the Jacobian of \(G(u)\).

More generally, this can be applied to implicit ODE and DAE for which the transient form is

\[F(u,\dot{u}) = 0. \]

For solving steady-state problems with pseudo-timestepping one proceeds as follows.

  • Provide the function G(u) with the routine

    TSSetRHSFunction(TS ts,Vec r,PetscErrorCode (*f)(TS,PetscReal,Vec,Vec,void*),void *fP);

    The arguments to the function f() are the timestep context, the current time, the input for the function, the output for the function and the (optional) user-provided context variable fP.

  • Provide the (approximate) Jacobian matrix of G(u) and a function to compute it at each Newton iteration. This is done with the command

    TSSetRHSJacobian(TS ts,Mat Amat, Mat Pmat,PetscErrorCode (*f)(TS,PetscReal,Vec,Mat,Mat,void*),void *fP);

    The arguments for the function f() are the timestep context, the current time, the location where the Jacobian is to be computed, the (approximate) Jacobian matrix, an alternative approximate Jacobian matrix used to construct the preconditioner, and the optional user-provided context, passed in as fP. The user must provide the Jacobian as a matrix; thus, if using a matrix-free approach, one must create a MATSHELL matrix.

In addition, the user must provide a routine that computes the pseudo-timestep. This is slightly different depending on if one is using a constant timestep over the entire grid, or it varies with location.

  • For location-independent pseudo-timestepping, one uses the routine

    TSPseudoSetTimeStep(TS ts,PetscInt(*dt)(TS,PetscReal*,void*),void* dtctx);

    The function dt is a user-provided function that computes the next pseudo-timestep. As a default one can use TSPseudoTimeStepDefault(TS,PetscReal*,void*) for dt. This routine updates the pseudo-timestep with one of two strategies: the default

    \[dt^{n} = dt_{\mathrm{increment}}*dt^{n-1}*\frac{|| F(u^{n-1}) ||}{|| F(u^{n})||} \]

    or, the alternative,

    \[dt^{n} = dt_{\mathrm{increment}}*dt^{0}*\frac{|| F(u^{0}) ||}{|| F(u^{n})||} \]

    which can be set with the call

    TSPseudoIncrementDtFromInitialDt(TS ts);

    or the option -ts_pseudo_increment_dt_from_initial_dt. The value \(dt_{\mathrm{increment}}\) is by default \(1.1\), but can be reset with the call

    TSPseudoSetTimeStepIncrement(TS ts,PetscReal inc);

    or the option -ts_pseudo_increment <inc>.

  • For location-dependent pseudo-timestepping, the interface function has not yet been created.

ARS97(1,2)

U.M. Ascher, S.J. Ruuth, and R.J. Spiteri. Implicit-explicit Runge-Kutta methods for time-dependent partial differential equations. Applied Numerical Mathematics, 25:151–167, 1997.

AP98

Uri M Ascher and Linda R Petzold. Computer methods for ordinary differential equations and differential-algebraic equations. Volume 61. SIAM, 1998.

BPR11

S. Boscarino, L. Pareschi, and G. Russo. Implicit-explicit Runge-Kutta schemes for hyperbolic systems and kinetic equations in the diffusion limit. Arxiv preprint arXiv:1110.4375, 2011.

ColomesB16

Oriol Colomés and Santiago Badia. Segregated Runge–Kutta methods for the incompressible Navier–Stokes equations. International Journal for Numerical Methods in Engineering, 105(5):372–400, 2016.

Con16(1,2)

E.M. Constantinescu. Estimating global errors in time stepping. ArXiv e-prints, March 2016. arXiv:1503.05166.

GKC13(1,2,3)

F.X. Giraldo, J.F. Kelly, and E.M. Constantinescu. Implicit-explicit formulations of a three-dimensional nonhydrostatic unified model of the atmosphere (NUMA). SIAM Journal on Scientific Computing, 35(5):B1162–B1194, 2013. doi:10.1137/120876034.

KC03(1,2,3)

C.A. Kennedy and M.H. Carpenter. Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math., 44(1-2):139–181, 2003. doi:10.1016/S0168-9274(02)00138-1.

PR05(1,2)

L. Pareschi and G. Russo. Implicit-explicit Runge-Kutta schemes and applications to hyperbolic systems with relaxation. Journal of Scientific Computing, 25(1):129–155, 2005.

RA05(1,2)

J. Rang and L. Angermann. New Rosenbrock W-methods of order 3 for partial differential algebraic equations of index 1. BIT Numerical Mathematics, 45(4):761–787, 2005.

SVB+97(1,2)

A. Sandu, J.G. Verwer, J.G. Blom, E.J. Spee, G.R. Carmichael, and F.A. Potra. Benchmarking stiff ode solvers for atmospheric chemistry problems II: Rosenbrock solvers. Atmospheric Environment, 31(20):3459–3472, 1997.