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High Order semi-implicit Rosenbrock type and Multistep methods for evolutionary partial differential equations with higher order derivatives

Boscarino Sebastiano, Giuseppe Izzo

TL;DR

The paper tackles the challenge of time integration for time-dependent PDEs with high-order spatial derivatives by developing semi-implicit Rosenbrock-type and partitioned semi-implicit multistep methods. By formulating the semi-discrete system as H(U,V) = F(U) + B(U)V and treating the stiff B(U)V term implicitly while keeping F(U) explicit, the authors obtain linearly implicit schemes that avoid Newton iterations and reduce restrictive Δt requirements. They derive order conditions for third-order SI-Rosenbrock schemes, propose a four-stage method with controlled stability through the parameter γ, and introduce modified SI-LM predictor–corrector schemes (SI-PC BDF$p$) to achieve higher-order accuracy with a fixed number of linear solves. Extensive 1D numerical experiments across diffusive, dispersive, and biharmonic-type PDEs confirm stability and convergence to the expected orders, using high-order finite-difference spatial discretizations and WENO for nonlinear convective terms. The work provides a versatile framework for high-order time discretization of complex PDEs, with clear pathways to extensions to higher dimensions and alternative spatial discretizations.

Abstract

The aim of this work is to apply a semi-implicit (SI) strategy within a Rosenbrock-type and IMEX linear multistep (LM) framework to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy provides great flexibility to treat these equations, and allows the construction of simple lienarly implicit schemes without any Newton iteration. Furthermore, the SI schemes so designed do not require the severe time-step restrictions typically encountered when using explicit methods for stability, i.e., $Δt = \mathcal{O}(Δx^k)$ for the $k$-th order PDEs with $k\ge 2$. For space discrertization, this strategy is combined with finite difference schemes. We provide example of methods up to order $p = 4$, and we illustrate the effectiveness of the schemes with appllications to dissipative, dispersive, and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and achieve the expected orders of accuracy.

High Order semi-implicit Rosenbrock type and Multistep methods for evolutionary partial differential equations with higher order derivatives

TL;DR

The paper tackles the challenge of time integration for time-dependent PDEs with high-order spatial derivatives by developing semi-implicit Rosenbrock-type and partitioned semi-implicit multistep methods. By formulating the semi-discrete system as H(U,V) = F(U) + B(U)V and treating the stiff B(U)V term implicitly while keeping F(U) explicit, the authors obtain linearly implicit schemes that avoid Newton iterations and reduce restrictive Δt requirements. They derive order conditions for third-order SI-Rosenbrock schemes, propose a four-stage method with controlled stability through the parameter γ, and introduce modified SI-LM predictor–corrector schemes (SI-PC BDF) to achieve higher-order accuracy with a fixed number of linear solves. Extensive 1D numerical experiments across diffusive, dispersive, and biharmonic-type PDEs confirm stability and convergence to the expected orders, using high-order finite-difference spatial discretizations and WENO for nonlinear convective terms. The work provides a versatile framework for high-order time discretization of complex PDEs, with clear pathways to extensions to higher dimensions and alternative spatial discretizations.

Abstract

The aim of this work is to apply a semi-implicit (SI) strategy within a Rosenbrock-type and IMEX linear multistep (LM) framework to a sequence of 1D time-dependent partial differential equations (PDEs) with high order spatial derivatives. This strategy provides great flexibility to treat these equations, and allows the construction of simple lienarly implicit schemes without any Newton iteration. Furthermore, the SI schemes so designed do not require the severe time-step restrictions typically encountered when using explicit methods for stability, i.e., for the -th order PDEs with . For space discrertization, this strategy is combined with finite difference schemes. We provide example of methods up to order , and we illustrate the effectiveness of the schemes with appllications to dissipative, dispersive, and biharmonic-type equations. Numerical experiments show that the proposed schemes are stable and achieve the expected orders of accuracy.
Paper Structure (9 sections, 1 theorem, 99 equations, 1 figure)

This paper contains 9 sections, 1 theorem, 99 equations, 1 figure.

Table of Contents

  1. Introduction
  2. SI methods
  3. SI Rosenbrock-type methods.
  4. Numerical Results.
  5. Partitioned semi-implicit Multistep methods.
  6. Semi-implicit predictor-corrector LM methods of order $p$
  7. Computational Aspects.
  8. Numerical Results
  9. Concluding remarks

Key Result

Proposition 4.1

Consider the differential equation (ydotzdot), with $\mathcal{H}(t,u,v)$ continuously differentiable and let $v(t) = u(t)$ be its solution. Assume condition LA and consider a SI-PC method Algor, where the predictor method (P) has order $p^*$ and the corrector one has order $p$. Then we have the foll

Figures (1)

  • Figure 1: Stability region (\ref{['Stab_region']}) for different values $\gamma$. Left $\gamma = 1-1/\sqrt{2}$, middle $\gamma = 3/4$, right $\gamma = 13/50$.

Theorems & Definitions (4)

  • Remark 3.1
  • Remark 4.1
  • Proposition 4.1
  • Example 4.1