We develop an alternative formulation of conservative finite difference weighted essentially non-oscillatory (WENO) schemes to solve conservation laws. In this formulation, the WENO interpolation of the solution and its derivatives are used to directly construct the numerical flux, instead of the usual practice of reconstructing the flux functions. Even though this formulation is more expensive than the standard formulation, it does have several advantages. The first advantage is that arbitrary monotone fluxes can be used in this framework, while the traditional practice of reconstructing flux functions can be applied only to smooth flux splitting. The second advantage, which is fully explored in this paper, is that a narrower effective stencil is used compared with previous high order finite difference WENO schemes based on the reconstruction of flux functions, with a Lax-Wendroff time discretization. We will describe the scheme formulation and present numerical tests for one- and two-dimensional scalar and system conservation laws demonstrating the designed high order accuracy and non-oscillatory performance of the schemes constructed in this paper.

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In earlier work, two of the authors constructed uniformly high order accurate finite volume and discontinuous Galerkin (DG) schemes satisfying a strict maximum principle for scalar conservation laws on rectangular meshes. The technique is generalized to positivity preserving (of density and pressure) high order DG or finite volume schemes for compressible Euler equations. The extension of these schemes to triangular meshes is conceptually plausible but highly nontrivial. In this paper, we first introduce a special quadrature rule which is exact for two-variable polynomials over a triangle of a given degree and satisfy a few other conditions, by which we can construct high order maximum principle satisfying finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or DG method solving two dimensional scalar conservation laws on triangular meshes. The same method can preserve the maximum principle for DG or finite volume schemes solving two-dimensional incompressible Euler equations in the vorticity stream-function formulation, or any passive convection equation with an incompressible velocity field. We also obtain positivity preserving (for density and pressure) high order DG or finite volume schemes solving compressible Euler equations on triangular meshes. Numerical tests for the third order Runge-Kutta DG (RKDG) method on unstructured meshes are reported.

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We present an efficient null space free Jacobi–Davidson method to compute the positive eigenvalues of time harmonic Maxwell’s equations. We focus on a class of spatial discretizations that guarantee the existence of discrete vector potentials, such as Yee’s scheme and the edge elements. During the Jacobi–Davidson iteration, the correction process is applied to the vector potential instead. The correction equation is solved approximately as in the standard Jacobi–Davidson approach. The computational cost of the transformation from the vector potential to the corrector is negligible. As a consequence, the expanding subspace automatically stays out of the null space and no extra projection step is needed. Numerical evidence confirms that the proposed scheme indeed outperforms the standard and projection-based Jacobi–Davidson methods by a significant margin.