Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Euler-Maruyama approximations of the stochastic heat equation on the sphere

Annika Lang, Ioanna Motschan-Armen

TL;DR

This work extends numerical analysis for SPDEs to the sphere by combining a spectral in space discretization with forward and backward Euler–Maruyama schemes in time for the stochastic heat equation on $\mathbb{S}^2$ driven by isotropic $Q$-Wiener noise. It derives optimal strong convergence rates that depend on the regularity of the initial data and the noise, and shows convergence of the mean and the second moment, with the second-moment rate roughly twice the strong rate. The approach leverages spherical harmonics and the Laplace–Beltrami operator to decouple modes into Ornstein–Uhlenbeck processes, enabling precise, element-wise error analysis without heavy semigroup machinery. Numerical experiments confirm the theoretical rates and demonstrate practical viability for sample-path simulations on the sphere.

Abstract

The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.

Euler-Maruyama approximations of the stochastic heat equation on the sphere

TL;DR

This work extends numerical analysis for SPDEs to the sphere by combining a spectral in space discretization with forward and backward Euler–Maruyama schemes in time for the stochastic heat equation on driven by isotropic -Wiener noise. It derives optimal strong convergence rates that depend on the regularity of the initial data and the noise, and shows convergence of the mean and the second moment, with the second-moment rate roughly twice the strong rate. The approach leverages spherical harmonics and the Laplace–Beltrami operator to decouple modes into Ornstein–Uhlenbeck processes, enabling precise, element-wise error analysis without heavy semigroup machinery. Numerical experiments confirm the theoretical rates and demonstrate practical viability for sample-path simulations on the sphere.

Abstract

The stochastic heat equation on the sphere driven by additive isotropic Wiener noise is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. Optimal strong convergence rates for a given regularity of the initial condition and driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown, where the approximation of the second moment converges with twice the strong rate. Numerical simulations confirm the theoretical results.
Paper Structure (7 sections, 7 theorems, 102 equations, 4 figures)

This paper contains 7 sections, 7 theorems, 102 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The stochastic heat equation on the sphere and its properties
  3. Spectral approximation in space
  4. Euler--Maruyama approximation in time
  5. Numerical simulation
  6. Properties of the solution
  7. Regularity of exponential functions and their approximation

Key Result

Lemma 3.1

Let $t \in \mathbb{T}$. Furthermore assume that there exist $\ell_0 \in {\mathbb N}$, $\alpha > 0$, and a constant $C>0$ such that the angular power spectrum $(A_\ell, \ell \in {\mathbb N}_0)$ satisfies $A_\ell \le C \cdot \ell^{-\alpha}$ for all $\ell > \ell_0$. Then the strong error of the approxi for all $\kappa \ge \ell_0$ and a constant $\hat{C}$ depending on $C$ and $\alpha$.

Figures (4)

  • Figure 1: Convergence of spectral approximation for different $\alpha$.
  • Figure 2: Convergence of the forward Euler--Maruyama scheme with respect to the time step size $h$ for different $\alpha$.
  • Figure 3: Convergence of the backward Euler--Maruyama scheme with respect to the time step size $h$ for different $\alpha$.
  • Figure 4: Strong convergence error based on $10$ Monte Carlo samples.

Theorems & Definitions (13)

  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Proposition 4.1
  • Proposition 4.2
  • Proposition 4.3
  • Theorem 4.4
  • proof
  • Theorem 4.5
  • proof
  • ...and 3 more