[Paper Review] High order schemes based on operator splitting and deferred corrections for stiff time dependent PDEs

Max Duarte, Matthew Emmett|arXiv (Cornell University)|Jul 1, 2014

Numerical methods for differential equations30 references3 citations

TL;DR

This paper presents a high-order time integration method for stiff time-dependent PDEs by combining operator splitting with iterative deferred corrections. It uses a low-order splitting solver to generate initial approximations at collocation nodes, which are then iteratively corrected using quadrature-based implicit Runge-Kutta schemes to achieve high-order accuracy while maintaining stability and efficiency.

ABSTRACT

We consider quadrature formulas of high order in time based on Radau-type, L-stable implicit Runge-Kutta schemes to solve time dependent stiff PDEs. Instead of solving a large nonlinear system of equations, we develop a method that performs iterative deferred corrections to compute the solution at the collocation nodes of the quadrature formulas. The numerical stability is guaranteed by a dedicated operator splitting technique that efficiently handles the stiffness of the PDEs and provides initial and intermediate solutions to the iterative scheme. In this way the low order approximations computed by a tailored splitting solver of low algorithmic complexity are iteratively corrected to obtain a high order solution based on a quadrature formula. The mathematical analysis of the numerical errors and local order of the method is carried out in a finite dimensional framework for a general semi-discrete problem, and a time-stepping strategy is conceived to control numerical errors related to the time integration. Numerical evidence confirms the theoretical findings and assesses the performance of the method in the case of a stiff reaction-diffusion equation.

Motivation & Objective

• To develop a high-order time integration scheme for stiff PDEs that avoids solving large nonlinear systems directly.
• To combine the stability of L-stable Radau–type implicit Runge–Kutta schemes with the efficiency of operator splitting.
• To achieve high-order accuracy through iterative deferred corrections applied to low-order splitting approximations.
• To ensure numerical stability and error control in the time integration of stiff semi-discrete PDEs.
• To provide a practical, low-complexity framework for high-order time integration suitable for large-scale PDE simulations.

Proposed method

• Uses $s$-stage $L$-stable Radau–type implicit Runge–Kutta schemes to define high-order quadrature formulas for time integration.
• Employs a dedicated operator splitting solver to compute low-order approximations at $s$ collocation nodes within each time step, ensuring stability and avoiding stiffness constraints.
• Applies iterative deferred corrections to improve the low-order splitting solutions toward the high-order quadrature solution at each collocation node.
• Relies on a time-stepping strategy with error control to manage local truncation errors in the iterative correction process.
• Defines the correction process such that the spectral integration operator $S_{t_{i-1}}^{t_i}({oldsymbol{U}})$ approximates the integral using quadrature nodes derived from the IRK scheme.
• Reinterprets the method as a variant of SDC (Spectral Deferred Correction) where the fine propagator is replaced by a splitting-based numerical integrator.

Experimental results

Research questions

• RQ1 Can high-order accuracy be achieved for stiff time-dependent PDEs without solving large nonlinear systems?
• RQ2 How can operator splitting be combined with deferred corrections to maintain stability while achieving high-order convergence?
• RQ3 What is the local order of accuracy and error behavior of the iterative correction process in the context of semi-discrete stiff PDEs?
• RQ4 How can effective error control be implemented in the time-stepping strategy for this method?
• RQ5 To what extent does the method outperform standard low-order splitting or fully coupled implicit Runge–Kutta schemes in terms of accuracy and efficiency?

Key findings

• The method achieves high-order accuracy (up to order $2s-1$ for $s$ stages) through iterative correction of low-order splitting solutions.
• Numerical experiments on a stiff reaction–diffusion equation confirm the theoretical order of accuracy and demonstrate optimal convergence rates.
• The iterative correction process exhibits superlinear convergence per iteration, with error reductions consistent with the theoretical analysis.
• The time-stepping strategy with error control successfully manages local truncation errors and maintains stability across stiff problems.
• The method maintains high efficiency by avoiding the solution of large nonlinear systems, relying instead on low-complexity splitting solves and iterative corrections.
• The approach is mathematically equivalent to a modified SDC scheme where the fine propagator is replaced by a splitting-based integrator, enabling stable high-order time integration.