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 timedependent PDEs.
Simple Example: Consider the PDE
discretized with centered finite differences in space yielding the semidiscrete equation
or with piecewise linear finite elements approximation in space \(u(x,t) \doteq \sum_i \xi_i(t) \phi_i(x)\) yielding the semidiscrete equation
Now applying the backward Euler method results in
in which
\(A\) 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
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 functionf()
are the timestep context, current time, input state \(u\), input time derivative \(\dot{u}\), and the (optional) userprovided contextfunP
. 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 semiimplicit (IMEX) method one also can provide an appropriate (approximate) Jacobian matrix of \(F()\).
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) userprovided contextjacP
.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 \(\dot{u} = (u^n  u^{n1})/\delta t\) 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) userprovided contextjacP
.
Providing appropriate \(F()\) and \(G()\) for your problem allows for the easy runtime switching between explicit, semiimplicit (IMEX), and fully implicit methods.
Basic TS Options#
The user first creates a TS
object with the command
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
TSEULER
, TSRK
(RungeKutta),
TSBEULER
, TSCN
(CrankNicolson), TSTHETA
, TSGLLE
(generalized linear), TSPSEUDO
, and TSSUNDIALS
(only if the
Sundials package is installed), or the command line optionts_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
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
The solve call implicitly sets up the timestep context; this can be done explicitly with
One destroys the context with
and views it with
TSView(TS ts,PetscViewer viewer);
In place of TSSolve()
, a single step can be taken using
DAE Formulations#
You can find a discussion of DAEs in [AP98] or Scholarpedia. In PETSc, TS deals with the semidiscrete 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
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 Index1 DAE#
This is a SemiExplicit Index1 DAE which has the form
where \(z\) is a new constraint variable, and the Jacobian \(\frac{dh}{dz}\) is nonsingular everywhere. We have suppressed the \(x\) dependence since it plays no role here. Using the nonsingularity 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 Index2 DAE#
This DAE has the form
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 index1 case. Our strategy will be to convert this into an index1 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,
If the Jacobian \(\frac{dh}{du} \frac{df}{dz}\) is nonsingular, then we have precisely a semiexplicit index1 DAE, and we can once again use the PETSc IMEX tools to solve it. A common example of an index2 DAE is the incompressible NavierStokes 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 RungeKutta method applied to this equation [ColomesB16].
Using ImplicitExplicit (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 nonoscillatory methods for hyperbolic PDE. The user provides three pieces of information, the APIs for which have been described above.
“Slow” part \(G(t,u)\) using
TSSetRHSFunction()
.“Stiff” part \(F(t,u,\dot u)\) using
TSSetIFunction()
.Jacobian \(F_u + \sigma F_{\dot u}\) using
TSSetIJacobian()
.
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()
) ora 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 RungeKutta, but can be solved by using fully implicit methods. Some usecase examples for TSARKIMEX
are listed in Table 13 and a list of methods with a summary of their properties is given in IMEX RungeKutta schemes.
\(\dot{u} = g(t,u)\) 
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 
\(\begin{aligned}F(t,u,\dot{u}) &= M \dot{u}  f(t,u) \\ G(t,u) &= 0\end{aligned}\) 
\(\dot{u} = f(t,u) + g(t,u)\) 
stiffnonstiff ODE 
\(\begin{aligned}F(t,u,\dot{u}) &= \dot{u}  f(t,u) \\ G(t,u) &= g(t,u)\end{aligned}\) 
\(M \dot{u} = f(t,u) + g(t,u)\) 
stiffnonstiff ODE with mass matrix 
\(\begin{aligned}F(t,u,\dot{u}) &= M\dot{u}  f(t,u) \\ G(t,u) &= M^{1} g(t,u)\end{aligned}\) 
\(\begin{aligned}\dot{u} &= f(t,u,z) + g(t,u,z)\\0 &= h(t,y,z)\end{aligned}\) 
semiexplicit index1 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 
Table 14 lists of the currently available IMEX RungeKutta schemes. For each method, it gives the ts_arkimex_type
name, the reference, the total number of stages/implicit stages, the order/stageorder, the implicit stability properties (IM), stiff accuracy (SA), the existence of an embedded scheme, and dense output (DO).
Name 
Reference 
Stages (IM) 
Order (Stage) 
IM 
SA 
Embed 
DO 
Remarks 

a2 
based on CN 
2 (1) 
2 (2) 
AStable 
yes 
yes (1) 
yes (2) 

l2 
SSP2(2,2,2) [PR05] 
2 (2) 
2 (1) 
LStable 
yes 
yes (1) 
yes (2) 
SSP SDIRK 
ars122 
ARS122 [ARS97] 
2 (1) 
3 (1) 
AStable 
yes 
yes (1) 
yes (2) 

2c 
[GKC13] 
3 (2) 
2 (2) 
LStable 
yes 
yes (1) 
yes (2) 
SDIRK 
2d 
[GKC13] 
3 (2) 
2 (2) 
LStable 
yes 
yes (1) 
yes (2) 
SDIRK 
2e 
[GKC13] 
3 (2) 
2 (2) 
LStable 
yes 
yes (1) 
yes (2) 
SDIRK 
prssp2 
PRS(3,3,2) [PR05] 
3 (3) 
3 (1) 
LStable 
yes 
no 
no 
SSP 
3 
[KC03] 
4 (3) 
3 (2) 
LStable 
yes 
yes (2) 
yes (2) 
SDIRK 
bpr3 
[BPR11] 
5 (4) 
3 (2) 
LStable 
yes 
no 
no 
SDIRK 
ars443 
[ARS97] 
5 (4) 
3 (1) 
LStable 
yes 
no 
no 
SDIRK 
4 
[KC03] 
6 (5) 
4 (2) 
LStable 
yes 
yes (3) 
yes 
SDIRK 
5 
[KC03] 
8 (7) 
5 (2) 
LStable 
yes 
yes (4) 
yes (3) 
SDIRK 
ROSW are linearized implicit RungeKutta methods known as Rosenbrock Wmethods. 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).
TS 
Reference 
Stages (IM) 
Order (Stage) 
IM 
SA 
Embed 
DO 
W 
PDAE 
Remarks 

theta1 
classical 
1(1) 
1(1) 
LStable 

theta2 
classical 
1(1) 
2(2) 
AStable 

2m 
Zoltan 
2(2) 
2(1) 
LStable 
No 
Yes(1) 
Yes(2) 
Yes 
No 
SSP 
2p 
Zoltan 
2(2) 
2(1) 
LStable 
No 
Yes(1) 
Yes(2) 
Yes 
No 
SSP 
ra3pw 
[RA05] 
3(3) 
3(1) 
AStable 
No 
Yes 
Yes(2) 
No 
Yes(3) 

ra34pw2 
[RA05] 
4(4) 
3(1) 
LStable 
Yes 
Yes 
Yes(3) 
Yes 
Yes(3) 

rodas3 
[SVB+97] 
4(4) 
3(1) 
LStable 
Yes 
Yes 
No 
No 
Yes 

sandu3 
[SVB+97] 
3(3) 
3(1) 
LStable 
Yes 
Yes 
Yes(2) 
No 
No 

assp3p3s1c 
unpub. 
3(2) 
3(1) 
AStable 
No 
Yes 
Yes(2) 
Yes 
No 
SSP 
lassp3p4s2c 
unpub. 
4(3) 
3(1) 
LStable 
No 
Yes 
Yes(3) 
Yes 
No 
SSP 
lassp3p4s2c 
unpub. 
4(3) 
3(1) 
LStable 
No 
Yes 
Yes(3) 
Yes 
No 
SSP 
ark3 
unpub. 
4(3) 
3(1) 
LStable 
No 
Yes 
Yes(3) 
Yes 
No 
IMEXRK 
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 vectorierr = 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 \(r1\))), 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].
TS 
Reference 
IM/EX 
\((p,r,s)\) 
\(\gamma\) 
Form 
Notes 



IM 
\((1,3,2)\) 
\(0.5\) 
\(y\varepsilon\) 
Based on backward Euler 

EX 
\((2,3,2)\) 
\(0\) 
\(y\varepsilon\) 


EX 
\((2,4,2)\) 
\(0\) 
\(y\tilde{y}\) 



EX 
\((2,5,2)\) 
\(0\) 
\(y\tilde{y}\) 


EX 
\((3,5,2)\) 
\(0\) 
\(y\tilde{y}\) 


EX 
\((2,6,2)\) 
\(0.25\) 
\(y\varepsilon\) 


EX 
\((3,8,2)\) 
\(0\) 
\(y\varepsilon\) 


EX 
\((2,9,2)\) 
\(0.25\) 
\(y\varepsilon\) 
Using fully implicit methods#
To use a fully implicit method like TSTHETA
, TSBDF
or TSDIRK
, 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 RungeKutta timestepper with variable timesteps#
The explicit Euler and RungeKutta methods require the ODE be in the form
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 RungeKutta timestepper with variable timesteps is an
implementation of the standard RungeKutta with an embedded method. The
error in each timestep is calculated using the solutions from the
RungeKutta method and its embedded method (the 2norm of the difference
is used). The default method is the \(3\)rdorder BogackiShampine
method with a \(2\)ndorder embedded method (TSRK3BS
). Other
available methods are the \(5\)thorder Fehlberg RK scheme with a
\(4\)thorder embedded method (TSRK5F
), the
\(5\)thorder DormandPrince RK scheme with a \(4\)thorder
embedded method (TSRK5DP
), the \(5\)thorder BogackiShampine
RK scheme with a \(4\)thorder embedded method (TSRK5BS
, and
the \(6\)th, \(7\)th, and \(8\)thorder robust Verner
RK schemes with a \(5\)th, \(6\)th, and \(7\)thorder
embedded method, respectively (TSRK6VR
, TSRK7VR
, TSRK8VR
).
Variable timesteps cannot be used with RK schemes that do not have an
embedded method (TSRK1FE
 \(1\)storder, \(1\)stage
forward Euler, TSRK2A
 \(2\)ndorder, \(2\)stage RK
scheme, TSRK3
 \(3\)rdorder, \(3\)stage RK scheme,
TSRK4
 \(4\)th order, \(4\)stage RK scheme).
Special Cases#
\(\dot{u} = A u.\) First compute the matrix \(A\) then call
TSSetProblemType(ts,TS_LINEAR); TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,NULL); TSSetRHSJacobian(ts,A,A,TSComputeRHSJacobianConstant,NULL);
or
TSSetProblemType(ts,TS_LINEAR); TSSetIFunction(ts,NULL,TSComputeIFunctionLinear,NULL); TSSetIJacobian(ts,A,A,TSComputeIJacobianConstant,NULL);
\(\dot{u} = A(t) u.\) Use
TSSetProblemType(ts,TS_LINEAR); TSSetRHSFunction(ts,NULL,TSComputeRHSFunctionLinear,NULL); TSSetRHSJacobian(ts,A,A,YourComputeRHSJacobian, &appctx);
where
YourComputeRHSJacobian()
is a function you provide that computes \(A\) as a function of time. Or useTSSetProblemType(ts,TS_LINEAR); TSSetIFunction(ts,NULL,TSComputeIFunctionLinear,NULL); TSSetIJacobian(ts,A,A,YourComputeIJacobian, &appctx);
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 ifTSSetSolutionFunction()
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 ifTSSetSolutionFunction()
is provided,TSMonitorDrawSolution()
.ts_monitor_solution binary[:filename]
 saves the solution at each iteration to a binary file,TSMonitorSolution()
. Solution viewers work with other timeaware formats, e.g.,ts_monitor_solution cgns:sol.cgns
, and can output one solution every 10 time steps by addingts_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 timestepping#
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.
ID 
Name 
Notes 


no adaptivity 


the default adaptor 


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., 
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 \(\widehat{p}=p1\) 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
and forms the acceptable error level
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
The error tolerances are satisfied when \(\rm wlte\le 1.0\).
The next step size is based on this error estimate, and determined by
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
zerocrossing, 1 for negative zerocrossing, and 0 for both),
terminate
conveys whether the timestepping should continue or halt
when an event is located, eventmonitor
is a user defined routine
that specifies the event description, postevent
is an optional
userdefined 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) userprovided 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) userprovided context eventP
.
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
(RungeKutta)
explicit methods and TSTHETA
implicit methods, which include
TSBEULER
and TSCN
.
Using the discrete adjoint methods#
Consider the ODE/DAE
and the cost function(s)
The TSAdjoint
routines of PETSc provide
and
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) userprovided 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 shortcircuits 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
where \(\phi\) is the phase angle and \(\omega\) is the frequency.
The initial conditions at time \(t_0\) are
\(p_{max}\) 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
with respect to the parameter \(p_m\). \(numcost\) is \(1\) since it is a scalar function.
For ODE solution, PETSc requires userprovided 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
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:
discrete adjoint sensitivity using explicit and implicit time stepping methods for an ODE problem TS Tutorial ex20adj,
an optimization problem using the discrete adjoint models of the ERK (for nonstiff ODEs) and the Theta methods (for stiff DAEs) TS Tutorial ex20opt_ic and TS Tutorial ex20opt_p,
an ODEconstrained optimization using the discrete adjoint models of the Theta methods for cost function with an integral term TS Power Grid Tutorial ex3opt,
discrete adjoint sensitivity using the CrankNicolson methods for DAEs with discontinuities TS Power Grid Stability Tutorial ex9busadj,
a DAEconstrained optimization problem using the discrete adjoint models of the CrankNicolson methods for cost function with an integral term TS Power Grid Tutorial ex9busopt,
discrete adjoint sensitivity using the CrankNicolson methods for a PDE problem TS AdvectionDiffusionReaction Tutorial ex5adj.
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 largescale
problems and longtime integration.
A socalled 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
andts_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 PETScSundials 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 downloadtchem
. 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 selfexplanatory 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
 7
Solving SteadyState Problems with PseudoTimestepping#
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 steadystate problem
we can use pseudotransient continuation by solving
Using time differencing
with the backward Euler method, we obtain nonlinear equations at a series of pseudotimesteps
For this problem the user must provide \(G(u)\), the time steps \(dt^{n}\) and the lefthandside 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
For solving steadystate problems with pseudotimestepping one proceeds as follows.
Provide the function
G(u)
with the routineTSSetRHSFunction(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) userprovided context variablefP
.Provide the (approximate) Jacobian matrix of
G(u)
and a function to compute it at each Newton iteration. This is done with the commandTSSetRHSJacobian(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 userprovided context, passed in asfP
. The user must provide the Jacobian as a matrix; thus, if using a matrixfree approach, one must create aMATSHELL
matrix.
In addition, the user must provide a routine that computes the pseudotimestep. This is slightly different depending on if one is using a constant timestep over the entire grid, or it varies with location.
For locationindependent pseudotimestepping, one uses the routine
TSPseudoSetTimeStep(TS ts,PetscInt(*dt)(TS,PetscReal*,void*),void* dtctx);
The function
dt
is a userprovided function that computes the next pseudotimestep. As a default one can useTSPseudoTimeStepDefault(TS,PetscReal*,void*)
fordt
. This routine updates the pseudotimestep with one of two strategies: the default\[dt^{n} = dt_{\mathrm{increment}}*dt^{n1}*\frac{ F(u^{n1}) }{ F(u^{n})} \]or, the alternative,
\[dt^{n} = dt_{\mathrm{increment}}*dt^{0}*\frac{ F(u^{0}) }{ F(u^{n})} \]which can be set with the call
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 callTSPseudoSetTimeStepIncrement(TS ts,PetscReal inc);
or the option
ts_pseudo_increment <inc>
.For locationdependent pseudotimestepping, the interface function has not yet been created.
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