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Operator splitting for abstract Cauchy problems. (English) Zbl 0913.65046
The splitting technique applied to the evolution problem of the form: \[ \tfrac{d}{dt}u=Au+f, \qquad t\in[0,T] \] with initial condition \[ u(0)=u_0 \] consists in
– dividing the interval \([0,T]\) on, say, \(N\) equal parts of the lenth \(h\) by points \(t_n=nh, \;n=0,1,\cdots N\),
– replacing the original problem on each interval \([t_n,t_{n+1}]\) by two simpler problems: \[ \tfrac d{dt}u_1=A_1u_1+f_1, \;u_1(t_{n+1})=u_2(t_n), \qquad \tfrac d{dt}u_2=A_2u_2+f_2, \] with \(A=A_1+A_2\). The author assumes that \(A\), \(A_1\) and \(A_2\) are infinitesimal generators of \(C_0\) semigroups with common domain. Two versions of conditions for splitting of \(f\) are discussed, one of them is simply \(f=f_1+f_2\). The main goal of this paper is a convergence theorem: \(u_2(t)\rightarrow u(t)\) when \(h\rightarrow 0\). This result is arranged in a very elegant way in the form of so called Lax theory: consistency (order) + stability implies convergence (with order). Consistency and stability questions are discussed in more detailed way. The author gives an example of an advection equation. Results of numerical experiments are presented.
A similar problem was discussed also by N. N. Yanenko [The method of fractional steps for solving multidimensional problems of mathematical physics (1967; Zbl 0183.18201)]. This kind of splitting method is called there the method of weak approximation, and only convergence problems, without stability and order questions, are discussed there.

65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
35K15 Initial value problems for second-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
65L05 Numerical methods for initial value problems
65L20 Stability and convergence of numerical methods for ordinary differential equations
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