Convergence analysis for a class of high-order semi-Lagrangian advection schemes. (English) Zbl 0914.65097

Falcone, Maurizio; Ferretti, Roberto

SIAM J. Numer. Anal. 35, No. 3, 909-940 (1998).

Following model problems are discussed: \[ v(x,t)_t=\lambda v(x,t)+f(x)\cdot\nabla v(x,t)+g(x);\quad v(x,0)=v_0(x) \]
\[ \lambda v(x,t)+ f(x)\cdot \nabla v(x,t)+ g(x)=0. \] The authors examine a class of semi-Lagrangian approximation schemes. In these methods the approximate solution is computed along a grid approximating the characteristics. The main results concern a priori estimates in \(L^\infty\) and \(L^2\) as well as the rate of convergence of the fully discrete scheme. By coupling time and space discretizations large time steps can be used without damaging the accuracy of the solutions. Results are illustrated by several numerical tests.

Reviewer: E.Schechter (Kaiserslautern)

MSC:

65M25	Numerical aspects of the method of characteristics for initial value and initial-boundary value problems involving PDEs
35L45	Initial value problems for first-order hyperbolic systems
65M12	Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs

Keywords:

hyperbolic equations; method of characteristics; semi-Lagrangian schemes; high-order schemes; convergence; stability; numerical tests