Convergence analysis of the splitting method to the nonlinear heat equation

TL;DR

This paper investigates the convergence properties of an operator splitting scheme applied to the nonlinear heat equation, establishing well-posedness, convergence rate, and supporting numerical evidence.

Contribution

The paper provides a rigorous convergence analysis of a splitting method for the nonlinear heat equation, including well-posedness and explicit convergence rate.

Findings

Established well-posedness of the approximation in L^r space

Derived convergence rate of order O(τ) for the splitting scheme

Numerical examples confirm theoretical results

Abstract

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ in $\Omega\times(0,\infty)$, $u=0$ in $\partial\Omega\times(0,\infty)$, $u ({\bf x},0) =\phi ({\bf x})$ in $\Omega$. where $\lambda\in\{-1,1\}$ and $\phi \in W^{1,q}(\Omega)\cap L^{\infty} (\Omega)$ with $2\leq p < \infty$ and $d(p-1)/2<q<\infty$. We establish the well-posedness of the approximation of $u$ in $L^r$-space ($r\geq q$), and furthermore, we derive its convergence rate of order $\mathcal{O}(\tau)$ for a time step $\tau>0$. Finally, we give some numerical examples to confirm the reliability of the analyzed result.