High-Order Unstructured Lagrangian One-Step WENO Finite Volume Schemes for Non-Conservative Hyperbolic Systems: Applications to Compressible Multi-Phase Flows

In this article we present the first better than second order accurate unstructured Lagrangian-type one-step WENO finite volume scheme for the solution of hyperbolic partial differential equations with non-conservative products. The method achieves high order of accuracy in space together with essentially non-oscillatory behavior using a nonlinear WENO reconstruction operator on unstructured triangular meshes. High order accuracy in time is obtained via a local Lagrangian space-time Galerkin predictor method that evolves the spatial reconstruction polynomials in time within each element. The final one-step finite volume scheme is derived by integration over a moving space-time control volume, where the non-conservative products are treated by a path-conservative approach that defines the jump terms on the element boundaries. The entire method is formulated as an Arbitrary-Lagrangian-Eulerian (ALE) method, where the mesh velocity can be chosen independently of the fluid velocity. The new scheme is applied to the full seven-equation Baer-Nunziato model of compressible multi-phase flows in two space dimensions. The use of a Lagrangian approach allows an excellent resolution of the solid contact and the resolution of jumps in the volume fraction. The high order of accuracy of the scheme in space and time is confirmed via a numerical convergence study. Finally, the proposed method is also applied to a reduced version of the compressible Baer-Nunziato model for the simulation of free surface water waves in moving domains. In particular, the phenomenon of sloshing is studied in a moving water tank and comparisons with experimental data are provided

Arbitrary-Lagrangian-Eulerian One-Step WENO Finite Volume Schemes on Unstructured Triangular Meshes

In this article we present a new class of high order accurate Arbitrary-Eulerian-Lagrangian (ALE) one-step WENO finite volume schemes for solving nonlinear hyperbolic systems of conservation laws on moving two dimensional unstructured triangular meshes. A WENO reconstruction algorithm is used to achieve high order accuracy in space and a high order one-step time discretization is achieved by using the local space-time Galerkin predictor. For that purpose, a new element--local weak formulation of the governing PDE is adopted on moving space--time elements. The space-time basis and test functions are obtained considering Lagrange interpolation polynomials passing through a predefined set of nodes. Moreover, a polynomial mapping defined by the same local space-time basis functions as the weak solution of the PDE is used to map the moving physical space-time element onto a space-time reference element. To maintain algorithmic simplicity, the final ALE one-step finite volume scheme uses moving triangular meshes with straight edges. This is possible in the ALE framework, which allows a local mesh velocity that is different from the local fluid velocity. We present numerical convergence rates for the schemes presented in this paper up to sixth order of accuracy in space and time and show some classical numerical test problems for the two-dimensional Euler equations of compressible gas dynamics.Comment: Accepted by "Communications in Computational Physics

Level Set Jet Schemes for Stiff Advection Equations: The SemiJet Method

Many interfacial phenomena in physical and biological systems are dominated by high order geometric quantities such as curvature. Here a semi-implicit method is combined with a level set jet scheme to handle stiff nonlinear advection problems. The new method offers an improvement over the semi-implicit gradient augmented level set method previously introduced by requiring only one smoothing step when updating the level set jet function while still preserving the underlying methods higher accuracy. Sample results demonstrate that accuracy is not sacrificed while strict time step restrictions can be avoided

High Order Maximum Principle Preserving Semi-Lagrangian Finite Difference WENO schemes for the Vlasov Equation

In this paper, we propose the parametrized maximum principle preserving (MPP) flux limiter, originally developed in [Z. Xu, Math. Comp., (2013), in press], to the semi- Lagrangian finite difference weighted essentially non-oscillatory scheme for solving the Vlasov equation. The MPP flux limiter is proved to maintain up to fourth order accuracy for the semi-Lagrangian finite difference scheme without any time step restriction. Numerical studies on the Vlasov-Poisson system demonstrate the performance of the proposed method and its ability in preserving the positivity of the probability distribution function while maintaining the high order accuracy