An important property for finite difference schemes designed on curvilinear meshes is the exact preservation of free-stream solutions. This property is difficult to fulfill for high order conservative essentially non-oscillatory (WENO) finite difference schemes. In this paper we explore an alternative flux formulation for such finite difference schemes which can preserve free-stream solutions, based on the numerical technique for the metric terms, which can be applied to this alternative flux formulation but is difficult to be applied to the standard finite difference formulation. Free-stream and vortex preservation properties are investigated, and comparison with standard finite difference WENO schemes is made. Theoretical derivation and numerical results show that the finite difference WENO schemes based on the alternative flux formulation can preserve free-stream and vortex solutions on both stationary and dynamically generalized coordinate systems, hence giving much better performance than the standard finite difference WENO schemes for such problems.

In this paper, we develop an efficient and energy stable fully-discrete local discontinuous Galerkin (LDG) method for the Cahn–Hilliard–Hele–Shaw (CHHS) system. The semi-discrete energy stability of the LDG method is proved firstly. Due to the strict time step restriction ( t = O( x4)) of explicit time discretization methods for stability, we introduce a semi- implicit time integration scheme which is based on a convex splitting of the discrete Cahn– Hilliard energy. The unconditional energy stability has been proved for this fully-discrete LDG scheme. The fully-discrete equations at the implicit time level are nonlinear. Thus, the nonlinear Full Approximation Scheme (FAS) multigrid method has been applied to solve this system of algebraic equations, which has been shown the nearly optimal complexity numerically. Numerical results are also given to illustrate the accuracy and capability of the LDG method coupled with the multigrid solver.

H(div) conforming and discontinuous Galerkin (DG) methods are designed for incompressible Euler's equation in two and three dimension. Error estimates are proved for both the semi-discrete method and fully-discrete method using backward Euler time stepping. Numerical examples exhibiting the performance of the methods are given.

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We propose a computational efficient yet simple numerical algorithm to solve the surface eikonal equation on general implicit surfaces. The method is developed based on the embedding idea and the fast sweeping methods. We first approximate the solution to the surface eikonal equation by the Euclidean weighted distance function defined in a tubular neighbourhood of the implicit surface, and then apply the efficient fast sweeping method to numerically compute the corresponding viscosity solution. Unlike some other embedding methods which require the radius of the computational tube satisfies h = O ( x ) for some h = O ( x ) , our approach allows h = O ( x ) . This implies that the total number of grid points in the computational tube is optimal and is given by h = O ( x ) for a co-dimensional one surface in h = O ( x ) . The method can be easily extended to general static HamiltonJacobi equation defined on implicit