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Second order explicit stabilized multirate method for stiff differential equations with error control

Mathieu Benninghoff, Gilles Vilmart

TL;DR

The paper tackles stiffness issues in semidiscrete parabolic problems with local mesh refinement by introducing a multirate explicit stabilized framework. It extends prior first-order multirate RKC to a second-order scheme, mROCK2, by constructing a second-order averaged force via two fast sub-solves and applying an $s$-stage ROCK2 macro-solve to the resulting modified equation, with error control and adaptive stepping. Theoretical contributions include stability and convergence analyses of the modified equation and the mROCK2 method, along with detailed stage-count and cost estimates. Numerical experiments on Robertson’s stiff problem, the heat equation with local refinement, and L-shaped diffusion demonstrate improved accuracy and efficiency, validating the approach for large-scale stiff systems with localized stiffness.”

Abstract

Explicit stabilized methods are highly efficient time integrators for large and stiff systems of ordinary differential equations especially when applied to semi-discrete parabolic problems. However, when local spatial mesh refinement is introduced, their efficiency decreases, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the system into fast stiff and slower mildly stiff components. In this context, [A. Abdulle, M.J. Grote and G. Rosilho de Souza 2022] proposed the order one multirate explicit stabilized method (mRKC). We extend their approach to second order and introduce the new multirate ROCK2 method (mROCK2), which achieves high precision and allows a step-size strategy with error control. Numerical methods including the heat equation with local spatial mesh refinements confirm the accuracy and efficiency of the scheme.

Second order explicit stabilized multirate method for stiff differential equations with error control

TL;DR

The paper tackles stiffness issues in semidiscrete parabolic problems with local mesh refinement by introducing a multirate explicit stabilized framework. It extends prior first-order multirate RKC to a second-order scheme, mROCK2, by constructing a second-order averaged force via two fast sub-solves and applying an -stage ROCK2 macro-solve to the resulting modified equation, with error control and adaptive stepping. Theoretical contributions include stability and convergence analyses of the modified equation and the mROCK2 method, along with detailed stage-count and cost estimates. Numerical experiments on Robertson’s stiff problem, the heat equation with local refinement, and L-shaped diffusion demonstrate improved accuracy and efficiency, validating the approach for large-scale stiff systems with localized stiffness.”

Abstract

Explicit stabilized methods are highly efficient time integrators for large and stiff systems of ordinary differential equations especially when applied to semi-discrete parabolic problems. However, when local spatial mesh refinement is introduced, their efficiency decreases, since the stiffness is driven by only the smallest mesh element. A natural approach is to split the system into fast stiff and slower mildly stiff components. In this context, [A. Abdulle, M.J. Grote and G. Rosilho de Souza 2022] proposed the order one multirate explicit stabilized method (mRKC). We extend their approach to second order and introduce the new multirate ROCK2 method (mROCK2), which achieves high precision and allows a step-size strategy with error control. Numerical methods including the heat equation with local spatial mesh refinements confirm the accuracy and efficiency of the scheme.
Paper Structure (22 sections, 10 theorems, 74 equations, 9 figures, 1 table, 2 algorithms)

This paper contains 22 sections, 10 theorems, 74 equations, 9 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Averaged force and modified equation
  3. Averaged force and limitations for acheiving second-order
  4. The second order averaged force
  5. Stability analysis of the modified equation
  6. Construction of explicit stabilized multirate methods
  7. The explicit stabilized Runge-Kutta methods
  8. The RKC method
  9. The ROCK2 method
  10. Explicit stabilized multirate Runge-Kutta method of order one (mRKC)
  11. New mROCK2 method: second-order multirate explicit stabilized explicit method
  12. New second-order mROCK2 method
  13. Variable time step with error control
  14. Computation of the number of stages
  15. Computational cost of the multirate ROCK2 method
  16. ...and 7 more sections

Key Result

Theorem 2.2

Let $\zeta<0$, then $\varphi(\eta\lambda)(\lambda+\zeta)(1-\frac{\eta\lambda}{2}\varphi(\eta\lambda)) \in [\frac{3}{2}\zeta,0]$ for all $\lambda\leq 0$, if $\eta\geq 2/|\zeta|$.

Figures (9)

  • Figure 1: The stability function for the RKC and the ROCK2 method.
  • Figure 2: mRKC on the multirate test equation with $\tau=1$, $\eta= \tfrac{6\tau}{\beta s^2} \tfrac{m^2}{m^2-1}$, $\varepsilon=0.05$, $s=10$, $m=10$, and $\zeta=\ell_s$.
  • Figure 3: Computational cost of the mROCK2 compared to the ROCK2 method
  • Figure 4: The inner and outer stability function of the mROCK2 for $w=\eta\zeta$, $z=\eta\lambda$, $\tau=1$, $\eta=\frac{2\tau(1+\alpha_m)}{\alpha_m\ell_s}$, $s=m=10$ and $w \approx 60$ chosen such that the condition $\gamma_\varepsilon\zeta \tau=0.81\ell_s$ for $\varepsilon=0.05$.
  • Figure 5: The inner stability function in the case of scale separation: $\tau \Phi_m(z)(z+w)(1-\frac{1}{2}z\alpha_m\Phi_m(z))$ for $w=\eta\zeta$, $z=\eta\lambda$, $\tau=1$, $\frac{2.8\tau}{\ell_s}$, $s=m=10$ and $w \approx 60$ chosen such that the condition $\gamma_\varepsilon\zeta \tau=0.81\ell_s$
  • ...and 4 more figures

Theorems & Definitions (22)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • Remark 4.3: Relaxed stability condition
  • Remark 4.4
  • Theorem 4.5
  • Lemma 4.6
  • proof
  • ...and 12 more