\(L_ 0\)-stable split linear multistep formulas for parabolic PDEs
Recently \textit{J. R. Cash} [SIAM J. Numer. Anal. 21, 433-446 (1984; Zbl 0553.65084)] investigated two new classes of finite difference schemes for the numerical solution of parabolic differential equations. The main idea is to exploit the important fact that all the extrapolation methods considered by \textit{A. R. Gourlay} and \textit{J. Ll. Morris} [ibid. 17, 641- 655 (1980; Zbl 0444.65058) and IMA J. Numer. Anal. 1, 347-357 (1981; Zbl 0482.65049)] for the numerical solution of the constant coefficient, homogeneous equation \(\delta u/\delta t=Lu\), \((x,t)\in <0,1>\times <0,T>\) subject to initial condition \(u(x,0)=f(x)\), \(x\in <0,1>\) and the boundary conditions \(u(0,t)=u(1,t)=0\) for \(t\geq 0\) can be written as semi-implicit Runge-Kutta methods. On this basis new types of split linear multistep methods are applied to the numerical solution of parabolic differential equations. The formulas derived achieve up to fourth-order in time, and unlike the Crank-Nicolson method, are stable in the space \(L_ 0\).
scientific article; zbMATH DE number 4147450