Convergence analysis of the splitting method to the nonlinear heat equation

Abstract

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\Omega \subset {\mathbb{R}}^d$ ($d\ge 1$): ${\partial }_{t}u=\Delta u+\lambda |u{|}^{p-1}u$ in $\Omega \times (0,\infty )$, $u=0$ in $\partial \Omega \times (0,\infty )$, $u(\mathbf{x},0)=\varphi (\mathbf{x})$ in $\Omega$. where $\lambda \in \{-1,1\}$ and $\varphi \in W^{1,q}(\Omega )\cap L^{\infty }(\Omega )$ with $2\le p<\infty$ and $d(p-1)/2<q<\infty$. We establish the well-posedness of the approximation of $u$ in $L^r$-space ($r\ge q$), 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.