Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An Exponential Stochastic Runge-Kutta Type Method of Order up to 1.5 for SPDEs of Nemytskii-type

Claudine von Hallern, Ricarda Mißfeldt, Andreas Rößler

TL;DR

This work develops ERKM1.5, a derivative-free exponential stochastic Runge-Kutta method for semilinear SPDEs with Nemytskii-type pointwise multiplicative noise. The scheme achieves strong temporal convergence of order up to $\gamma\in[1,\tfrac{3}{2})$ while keeping computational cost near linear in the problem dimension, contrasting with derivative-heavy Taylor-type methods. The authors provide a rigorous framework, a uniform $L^p$ moment bound, and a complete convergence proof, supported by spectral Galerkin implementation and numerical experiments including explicit, linear multiplicative, and nonlinear examples. The results demonstrate that ERKM1.5 offers a practical and efficient avenue for high-order SPDE simulations with reduced derivative evaluation burden and favorable scaling in dimensionality.

Abstract

For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge-Kutta type that allows for convergence with a temporal order of up to 3/2 and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

An Exponential Stochastic Runge-Kutta Type Method of Order up to 1.5 for SPDEs of Nemytskii-type

TL;DR

This work develops ERKM1.5, a derivative-free exponential stochastic Runge-Kutta method for semilinear SPDEs with Nemytskii-type pointwise multiplicative noise. The scheme achieves strong temporal convergence of order up to while keeping computational cost near linear in the problem dimension, contrasting with derivative-heavy Taylor-type methods. The authors provide a rigorous framework, a uniform moment bound, and a complete convergence proof, supported by spectral Galerkin implementation and numerical experiments including explicit, linear multiplicative, and nonlinear examples. The results demonstrate that ERKM1.5 offers a practical and efficient avenue for high-order SPDE simulations with reduced derivative evaluation burden and favorable scaling in dimensionality.

Abstract

For the approximation of solutions for stochastic partial differential equations, numerical methods that obtain a high order of convergence and at the same time involve reasonable computational cost are of particular interest. We therefore propose a new numerical method of exponential stochastic Runge-Kutta type that allows for convergence with a temporal order of up to 3/2 and that can be combined with several spatial discretizations. The developed family of derivative-free schemes is tailored to stochastic partial differential equations of Nemytskii-type, i.e., with pointwise multiplicative noise operators. We prove the strong convergence of these schemes in the root mean-square sense and present some numerical examples that reveal the theoretical results.

Paper Structure

This paper contains 20 sections, 6 theorems, 104 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The exponential stochastic Runge-Kutta type method
  3. Setting
  4. A family of exponential stochastic Runge-Kutta type schemes and a convergence result
  5. Computational cost of the exponential stochastic Runge-Kutta type schemes
  6. Implementation and numerical examples
  7. Spectral Galerkin implementation
  8. Simulation of mixed integrals
  9. Numerical examples
  10. Example with an explicitly given solution
  11. Stochastic heat equation with linear multiplicative noise
  12. A nonlinear example
  13. Proofs
  14. Proof of a uniform bound for approximations
  15. Preliminaries
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let some arbitrary coefficients ${c_1}, {c_2}, {c_3}, {c_4}, {c_5}, {c_6}, {c_7} \in \mathbb{R} \setminus \{0\}$ be given, let $p \geq 2$ and assume that for some constants $\gamma\in[1,\frac{3}{2})$, $\alpha \in(\gamma-1,\gamma]$, $\beta \in (\gamma -\tfrac{1}{2}, \gamma]$, $\delta \in (\gamma-1, \beta]$, $\theta = \max \{ \gamma-\alpha, \gamma-\tfrac{1}{2}, 2(\gamma-\beta), 2(\gamma-\delta-\

Figures (3)

  • Figure 1: $L^2(\Omega,H)$-error at time $T=1$ versus number of time steps in $\log$-$\log$-scale for $M = 2^l$, $l\in\{2,3,\cdots,16\}$, for Example 1 in Section \ref{['Sec:ExactSolution']}.
  • Figure 2: $L^2(\Omega,H)$-error at $T=1$ versus number of time steps for $M = 2^l$, $l\in\{2,3,\cdots,14\}$ for Example 2 in Section \ref{['Sec:Example2']}.
  • Figure 3: $L^2(\Omega,H)$-error at $T=1$ versus number of time steps $M = 2^l$, $l\in\{2,3,\cdots,14\}$ for Example 3 in Section \ref{['Sec:Example3']}.

Theorems & Definitions (13)

  • Proposition 2.1
  • Theorem 2.2
  • Lemma 3.1
  • Remark 3.1
  • proof : Proof of Proposition \ref{['Prop:Bounds']}
  • proof : Proof of Theorem \ref{['Prop:Conv']}
  • Lemma 4.1
  • proof
  • Remark 4.1
  • Lemma 4.2
  • ...and 3 more