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High order conservative Lagrangian schemes with Lax-Wendroff type time discretization for the compressible Euler equations. (English) Zbl 1287.76181
Summary: We explore the Lax-Wendroff (LW) type time discretization as an alternative procedure to the high order Runge-Kutta time discretization adopted for the high order essentially non-oscillatory (ENO) Lagrangian schemes developed in (*) [J. Cheng and Ch.-W. Shu, J. Comput. Phys. 227, No. 2, 1567–1596 (2007; Zbl 1126.76035)] and (**) [Commun. Comput. Phys. 4, 1008–1024 (2008)]. The LW time discretization is based on a Taylor expansion in time, coupled with a local Cauchy-Kowalewski procedure to utilize the partial differential equation (PDE) repeatedly to convert all time derivatives to spatial derivatives, and then to discretize these spatial derivatives based on high order ENO reconstruction. Extensive numerical examples are presented, for both the second-order spatial discretization using quadrilateral meshes (*) and third-order spatial discretization using curvilinear meshes (**). Comparing with the Runge-Kutta time discretization procedure, an advantage of the LW time discretization is the apparent saving in computational cost and memory requirement, at least for the two-dimensional Euler equations that we have used in the numerical tests.

MSC:
76M20 Finite difference methods applied to problems in fluid mechanics
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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