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 $\Omega\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$ 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.