Trapezoidal and midpoint splittings for initial-boundary value problems

In this paper we consider various multi-component splittings based on the trapezoidal rule and the implicit midpoint rule. It will be shown that an important requirement on such methods is internal stability. The methods will be applied to initial-boundary value problems. Along with a theoretical analysis, some numerical test results will be presented.

[1]  Alexander Ostermann,et al.  Interior estimates for time discretizations of parabolic equations , 1995 .

[2]  G. Marchuk.  Splitting and alternating direction methods , 1990 .

[3]  Randall J. LeVeque,et al.  Intermediate boundary conditions for LOD, ADI and approximate factorization methods , 1985 .

[4]  J. Kraaijevanger,et al.  B-convergence of the implicit midpoint rule and the trapezoidal rule , 1985 .

[5]  N. N. Yanenko,et al.  The Method of Fractional Steps , 1971 .

[6]  Ernst Hairer,et al.  Solving Ordinary Differential Equations I: Nonstiff Problems , 2009 .

[7]  N. N. I︠A︡nenko.  The Method of Fractional Steps: The Solution of Problems of Mathematical Physics in Several Variables , 1971 .

[8]  Raimondas Čiegis,et al.  On the stability of LOD difference schemes with respect to boundary conditions , 1994 .

[9]  Willem Hundsdorfer.  Unconditional convergence of some Crank-Nicolson LOD methods for initial-boundary value problems , 1990 .

[10]  Willem Hundsdorfer,et al.  A note on stability of the Douglas splitting method , 1998, Math. Comput..

[11]  V. Thomée,et al.  Single step methods for inhomogeneous linear differential equations in Banach space , 1982 .

[12]  R. F. Warming,et al.  An implicit finite-difference algorithm for hyperbolic systems in conservation-law form. [application to Eulerian gasdynamic equations , 1976 .

[13]  Willem Hundsdorfer,et al.  Stability and convergence of the Peaceman-Rachford ADI method for initial-boundary value problems , 1987 .