Abstract In [22], 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 in [23]. The extension...

We construct uniformly high order accurate schemes satisfying a strict maximum principle for scalar conservation laws. A general framework (for arbitrary order of accuracy) is established to construct a limiter for finite volume schemes (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO) schemes) or discontinuous Galerkin (DG) method with first order Euler forward time discretization...

In [16, 17], we constructed uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for the Euler equations of compressible gas dynamics with the ideal gas equation of state. The technique also applies to high order accurate finite volume schemes. For the Euler equations with various source terms (e.g., gravity and chemical reactions),...

In this work, we discuss a family of finite element discretizations for the incompressible Stokes problem using continuous pressure approximations on simplicial meshes. We show that after a simple and cheap correction, the mass-fluxes obtained by the considered schemes preserve local conservation on dual cells without reducing the convergence order. This allows the direct coupling to vertex-cen...

In the last few years, two new high-order accurate methods for unstructured grids have been developed, namely the spectral volume (SV) and the spectral difference (SD) method. Both methods are related to the well-known discontinuous Galerkin (DG) method, see for instance [1], in the sense that they also use piecewise continuous polynomials as solution approximation space. The development of the...

In the last decades, Discontinuous Galerkin (DG) methods have seen rapid growth and are widely used in various application domains (see [13] for an historical introduction). This is due to their main advantage of combining the best of finite element and finite volume methods. For the time-harmonic Maxwell equations, once the problem is discretized with a DG method, finding robust solvers is a d...

The Runge–Kutta discontinuous Galerkin (RKDG) method for solving hyperbolic conservation laws is a high order finite element method, which utilizes the useful features from high resolution finite volume schemes, such as the exact or approximate Riemann solvers, TVD Runge–Kutta time discretizations, and limiters. In this paper, we investigate using the RKDG finite element method for compressible...

Stochastic Galerkin finite element discretisations of PDEs with stochastically nonlinear coefficients lead to linear systems of equations with block dense matrices. In contrast, stochastic Galerkin finite element discretisations of PDEs with stochastically linear coefficients lead to linear systems of equations with block sparse matrices which are cheaper to manipulate and precondition in the f...

A stable volume integral equation formulation based on equivalent volumetric currents is presented for modeling electromagnetic scattering of highly inhomogeneous dielectric objects. The proposed formulation is numerically solved by means of Galerkin method of moments on uniform grids, allowing for acceleration of the matrix-vector products associated with the iterative solver with the help of ...