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

65M25Method of characteristics (IVP of PDE, numerical methods)
35L45First order hyperbolic systems, initial value problems
65M12Stability and convergence of numerical methods (IVP of PDE)
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