Convergence analysis of the splitting method to the nonlinear heat equation

In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $\mathrm{\Omega }\subset {\mathbb{R}}^d$ ($d\ge 1$): $\partial_t u = \Delta u + \lambda |u|^{p-1} u$in$\Omega\times(0,\infty)$,$u=0$in$\partial\Omega\times(0,\infty)$,$u ({\bf x},0) =\phi ({\bf x})$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 ($r\ge q$), 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.