An unconditionally stable and \(O(\tau^2+h^4)\) order \(L_\infty\) convergent difference scheme for linear parabolic equations with variable coefficients (Q2762700)

The author applies a finite difference scheme of \textit{M. K. Jain}, \textit{R. K. Jain} and \textit{R. K. Mohanty} [Numer. Methods Partial Differ. Equations 6, No. 4, 311-319 (1990; Zbl 0715.65067)] firstly to the following problem NEWLINE\[NEWLINEr(x,t) u_t- u_{xx}= f(x, t),\quad 0< x< 1,\quad 0\leq t< T,NEWLINE\]NEWLINE NEWLINE\[NEWLINEu(x,0)= \varphi(x),\quad 0< x< 1,\quad u(0,t)= u(1,t)= 0,\quad 0\leq t< T,NEWLINE\]NEWLINE where \(r(x,t)\geq c_0> 0\), \(\varphi(0)= \varphi(1)= 0\), and next to the following one NEWLINE\[NEWLINEr(x,t) u_t- u_{xx}- b(x,t) u_x+ c(x, t)u= f(x,t).NEWLINE\]NEWLINE He proves that the difference method is unconditionally stable and convergent with order \(O(\tau^2+ h^4)\) (\(\tau\) and \(h\) are sizes of time and space steps, respectively) in the \(L_\infty\)-norm. He shows that the method works well also for complex \(r\) and \(f\), when \(\text{Re }r(x,t)\geq c_0> 0\).