On operator splitting for unsteady boundary value problems (Q1093331)

An initial value problem is considered for the equation \[ \partial u/\partial t=(\partial F/\partial x)+(\partial G/\partial y)+H,\quad t>0,\quad -\infty <x,y<\infty, \] F(u)\(=f_ 1(u)+f_ 2(u,\partial u/\partial x)\), \(G(u)=g_ 1(u)+g_ 2(u,\partial u/\partial x)\) which is assumed to have for \(0\leq t\leq T\) a unique solution in the class \(C^{p+1}\), for some \(p\geq 2\). Second order splittings are obtained, which generalize previous results of \textit{R. W. MacCormack} [Lecture Notes Phys. 8, 151-163 (1971; Zbl 0228.76098)].