论文标题
分裂方法与非线性热方程的收敛分析
Convergence analysis of the splitting method to the nonlinear heat equation
论文作者
论文摘要
在本文中，我们分析了$ω\ subset \ mathbb {r}^d $（$ d \ geq 1 $）中的非线性热方程的操作员分裂方案：$ \ partial_t u =ΔU=Δu +Δu +λ| $ \partialΩ\ times（0，\ infty）$，$ u（{\ bf x}，0）= ϕ（{\ bf x}）$ in $ω$。其中$λ\ in \ { - 1,1 \} $和$ nev {1，q}（ω）\ cap l^{\ infty}（ω）$，带有$ 2 \ leq p <\ iffty $和$ d（p-1）和$ d（p-1）/2 <q <q <q <\ iffty $。我们建立了$ u $ in $ l^r $ -space（$ r \ geq q $）的近似值的良好性，此外，我们得出了其订单$ \ nathcal {o}（τ）$的收敛速率，以$ \ thime $τ）$ 0 $ 0 $。最后，我们给出一些数值示例，以确认分析结果的可靠性。
In this paper, we analyze an operator splitting scheme of the nonlinear heat equation in $Ω\subset\mathbb{R}^d$ ($d\geq 1$): $\partial_t u = Δu + λ|u|^{p-1} u$ in $Ω\times(0,\infty)$, $u=0$ in $\partialΩ\times(0,\infty)$, $u ({\bf x},0) =ϕ({\bf x})$ in $Ω$. where $λ\in\{-1,1\}$ and $ϕ\in W^{1,q}(Ω)\cap L^{\infty} (Ω)$ 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 $\mathcal{O}(τ)$ for a time step $τ>0$. Finally, we give some numerical examples to confirm the reliability of the analyzed result.
