Time Integration Project: Robust time stepping for stiff and multiscale PDEs

This project develops time integration algorithms for stiff and multiscale dynamical systems arising from PDE discretizations. The focus is on methods that remain accurate in long simulations while scaling to large problems: strong-stability-preserving (SSP) schemes, general linear methods (GLMs), implicit-explicit (IMEX) integration, multirate time stepping, and global error estimation.

Thrust: Time integration Scope: SSP, GLM, IMEX, multirate, global error estimation Implementation: PETSc TS and DESolve

Scientific question

How should time integrators adapt to stiff, multiscale dynamics?

After spatial discretization, many PDEs become very large ODE or DAE systems whose fastest scales can dominate stability, cost, and accumulated error.

Core idea

Exploit structure instead of brute force

The methods here use SSP design, implicit-explicit splitting, multirate partitions, and explicit global-error tracking to target the right source of difficulty in each problem.

Practical outcome

Algorithms that land in production software

A key objective is not just analysis, but methods that can be implemented in libraries such as PETSc TS and DESolve for real scientific simulation workflows.

Motivation

Many scientific applications evolve PDEs in time. After spatial discretization, these problems typically take the form of large ODE systems:

\[ \frac{d u_h(t)}{dt} = F_h(u_h(t), t), \qquad u_h(0) = u_{h,0}. \]

Two challenges tend to dominate:

  1. Stiffness: fast physics or fine spatial scales can force prohibitively small explicit time steps.
  2. Multiple time scales: different components (or spatial regions) evolve at very different rates.

This project explores integrators that exploit structure (splittings, partitions, and error estimators) to improve efficiency while maintaining stability and reliable accuracy.

Topic 1: New methodologies for numerical integrators

At the core are one-step (Runge-Kutta) and multistep (linear multistep) methods. For hyperbolic PDEs and nonlinear conservation laws, robustness often means preserving stability properties that are known to hold under forward Euler, such as monotonicity or total-variation-diminishing (TVD) behavior.

Strong stability preserving (SSP) time stepping

An SSP method is designed so that, under an appropriate step-size restriction, it can be expressed as a convex combination of forward Euler updates. This provides a principled way to build higher-order methods that inherit nonlinear stability properties important in PDE simulations.

General linear methods (GLMs)

GLMs unify Runge-Kutta and linear multistep methods within a single framework. A compact representation is:

\[ Y = h A F(Y) + U\,y^{[n-1]}, \qquad y^{[n]} = h B F(Y) + V\,y^{[n-1]}, \]

where $Y$ collects stage values, $y^{[n]}$ stores the propagated solution history, and $(A,B,U,V)$ define the method. This view is useful for designing schemes with properties that matter in PDE settings, such as high stage order (to mitigate order reduction in stiff problems) and SSP behavior. GLMs also provide a natural bridge to linearly implicit (Rosenbrock-type) constructions and to methods with embedded error estimation.

Advantages and limitations

Advantages: principled stability design (SSP) and a unified design space (GLMs) that can improve long-time robustness for PDE discretizations.

Limitations: higher-order SSP/GLM designs can increase stage count or coupling complexity, and practical performance depends on implementation details (e.g., Jacobian reuse, preconditioning, and adaptivity).

Selected references

Topic 2: Implicit-explicit (IMEX) and splitting methods

In many PDE models, different terms have very different stiffness and solver cost. IMEX methods start from an additive split

\[ F_h(u,t) = F_E(u,t) + F_I(u,t), \]

where $F_E$ is treated explicitly (cheap, nonstiff) and $F_I$ is treated implicitly (stiff, but often structured so that robust solvers and preconditioners apply). IMEX is particularly attractive for multiphysics coupling and for systems where a fully implicit method would be too expensive.

When IMEX works best

Best use cases: a clean split with a stiff component that is well-handled by scalable implicit solvers, and a nonstiff component that would otherwise enforce small explicit steps.

Typical limitations: splitting error and nonlinear coupling can reduce effective order (order reduction), and stability constraints may still depend on the explicit component or on how the split is defined.

Accuracy and cost tradeoffs in IMEX designs

IMEX design is ultimately about balancing stability and accuracy against solver cost. This figure shows a representative cost-versus-error comparison across different IMEX-type strategies.

Error versus CPU time comparison across IMEX and related methods
Error versus CPU time illustrates efficiency tradeoffs among IMEX variants.

Selected references

Topic 3: Multirate methods for multiple time scales

Multirate methods target problems where some components evolve fast and others evolve slowly. One way to express this is a partition:

\[ \frac{d u}{dt} = f_{\mathrm{slow}}(u,t) + f_{\mathrm{fast}}(u,t). \]

The goal is to take a large macro step for the slow dynamics while resolving the fast dynamics with cheaper micro steps, often only where needed (e.g., localized fast regions in space).

This project studies multirate methods built from several design patterns:

  1. Multirate Runge-Kutta (MRK): RK stages are coupled across fast/slow components with controlled information exchange.
  2. Multirate linear multistep (MR-LMM): multistep history is advanced with different step sizes for different components.
  3. Multirate extrapolation: a hierarchy of solves at different resolutions is combined by extrapolation to improve accuracy without fully resolving all components at the smallest step size.

Fast and slow regions, coupling, and stability

The central multirate question is how to couple fast and slow updates so that the combined method is stable and accurate. The figures below illustrate fast/slow structure, error behavior, and the impact of coupling choices on stability.

Example of a spatially varying wavespeed with fast and slow regions
Spatially varying fast/slow structure motivates multirate stepping.
Error behavior for system, fast, and slow components as a function of time step
Different components can exhibit different error sensitivity to the step size.
An unstable multirate integration example with blow-up behavior
Naive coupling can produce severe instabilities in multirate settings.
A stable multirate integration example
Structured coupling stabilizes multirate integration in the same setting.

Advantages and limitations

Advantages: reduces cost by resolving fast dynamics only as needed while taking large steps for slow dynamics.

Limitations: stability and accuracy depend on coupling design; fast/slow partitions that change in time or space require careful treatment to avoid drift or instability.

Selected references

Topic 4: Global error estimation and control

Most adaptive time integrators choose the time step by controlling a local error estimate (LEE), for example from an embedded method pair. This is effective for short integrations, but it does not directly control the global error $e^{[n]} = y^{[n]} - y(t_n)$, which can drift above tolerance over long horizons or in problems with unstable components.

Global error estimation methods augment the integrator so that an estimate of the accumulated error is propagated alongside the solution. In the approach developed in this project, a general linear method evolves both the numerical solution $y^{[n]}$ and an error estimate $\varepsilon^{[n]}$:

\[ \{y^{[n]},\varepsilon^{[n]}\} = \mathcal{G}_h(\{y^{[n-1]},\varepsilon^{[n-1]}\}), \qquad \widehat{y}^{[n]} = y^{[n]} + \varepsilon^{[n]}, \qquad \varepsilon_{\mathrm{loc}}^{[n]} = \varepsilon^{[n]} - \varepsilon^{[n-1]}. \]

Advantage. The estimator targets the quantity of interest (global error) rather than an indirect proxy, which is better suited for long-time integration and for detecting when local-error-controlled simulations silently accumulate unacceptable error.

Limitation. As with any estimator, accuracy depends on the problem class and method design; global-error-aware adaptivity introduces additional parameters and requires careful stability analysis.

Local versus global error behavior

Local error control can appear successful on short windows while the accumulated global error grows over long horizons. Global error estimation explicitly tracks the accumulated error and can remain informative in long-time simulations.

Global error over a short time interval under local error control
Short time: local error control can keep the global error below tolerance.
Global error growth over a long time interval under local error control
Long time: global error can grow beyond tolerance even when local error is controlled.
Estimated global error compared with exact global error
Estimated versus exact global error illustrates estimator fidelity.
Stability diagnostics for global error estimation methods
Stability diagnostics complement accuracy when designing global-error-aware schemes.

Selected references

Software

Many of the methods above are available in open-source libraries used in production simulations.

PETSc TS

HPC time stepping

Includes ARKIMEX, global error estimation methods (GLEE), and multirate time stepping for large-scale ODE/DAE systems.

Project page

DESolve

Time integration package

Research implementation of MREXIM, along with prototyping of stiff and multirate integrators.

Website GitHub

Funding

References for deeper dive

The references below are selected to match the four method families highlighted on this page.