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.
Categories: math.NA cs.NA math.AP