Uniform convergence of the implicit scheme of the finite-difference method for solving the first boundary-value problem for a nonlinear second-order parabolic equation

M. N. Yakovlev

Zapiski Nauchnykh Seminarov POMI, Computational methods and algorithms, Tome 70 (1977), pp. 241-255

Résumé

Let $u(t,x)$ be a solution of the first initial–boundary-value problem for the quasilinear parabolic equation \begin{equation} \dfrac{\partial u}{\partial t}=a(t,x,u,\dfrac{\partial u}{\partial x})\dfrac{\partial^2u}{\partial x^2}+ b(t,x,u,\dfrac{\partial u}{\partial x}),\qquad 0<t\leqslant T,\quad 0<x<1 \tag{1} \end{equation} with the initial condition \begin{equation} u(0,x)=\omega(x),\quad 0<x<1 \tag{2} \end{equation} and the boundary conditions \begin{equation} u(t,0)=u(t,1)=0,\quad 0<t\leqslant T, \tag{3} \end{equation} such that $$ \biggl|\dfrac{\partial^4u}{\partial x^4}(t,x)\biggr|\leqslant C,\quad \biggl|\dfrac{\partial^2u}{\partial t^2}(t,x)\biggr|\leqslant\dfrac{c}{t^\sigma},\quad 0\leqslant\sigma<2 $$ Assume that the functions $a(t,x,u,p)$, $b(t,x,u,p)$ are smooth and in a small neighborhood of the solution under consideration. Then, the implicit scheme of the finite-difference method converges uniformly to the solution under consideration with the order$h^2+\varphi(\tau)$, under the condition that \begin{equation} \varphi(\tau)\leqslant\beta h^\gamma,\quad \beta>0,\quad \gamma>1 \tag{4} \end{equation} Here $$ \varphi(\tau)= \begin{cases} \tau & \text{\rm{ при }}0\leqslant\sigma<1\\ \tau\ln\dfrac{T}{\tau} & \text{\rm{ при }}\sigma=1\\ \tau^{2-\sigma} & \text{\rm{ при }}1<\sigma<2. \end{cases} $$ One also considers convergence conditions when the relations (4) do not hold, convergence conditions for equations of the form $$ \dfrac{\partial u}{\partial t}=F\biggl(t,x,u,\dfrac{\partial u}{\partial x},\dfrac{d}{dx}K(t,x,\dfrac{\partial u}{\partial x})\biggr) $$ and weakly connected systems of such equations.