Operator splitting for abstract Cauchy problems.

*(English)*Zbl 0913.65046The 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.

– 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.

Reviewer: K.Moszyński (Warszawa)

##### MSC:

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 |