To simulate the rapid transient streamer discharge process, a method with high resolution and high order accuracy is usually required. This paper presents such a method based on discontinuous Galerkin (DG) method combined with hierarchical reconstruction (HR) for simulating the fluid model of streamer discharges. Numerical comparisons suggest that DG with HR is superior in both resolution and a...

We construct a local Lax-Friedrichs type positivity-preserving flux for compressible Navier-Stokes equations, which can be easily extended to high dimensions for generic forms of equations of state, shear stress tensor and heat flux. With this positivity-preserving flux, any finite volume type schemes including discontinuous Galerkin (DG) schemes with strong stability preserving Runge-Kutta tim...

Robust and accurate schemes are designed to simulate the coupling between subsurface and overland flows. The coupling conditions at the interface enforce the continuity of both the normal flux and the pressure. Richards’ equation governing the subsurface flow is discretized using a Backward Differentiation Formula and a symmetric interior penalty Discontinuous Galerkin method. The kinematic wav...

The entropy solutions of the compressible Euler equations satisfy a minimum principle for the specific entropy [11]. First order schemes such as Godunov-type and Lax-Friedrichs schemes and the second order kinetic schemes [6] also satisfy a discrete minimum entropy principle. In this paper, we show an extension of the positivity-preserving high order schemes for the compressible Euler equations...

The Meshless Finite Volume Method (MFVM) is developed for solving elasto-static problems, through a new Meshless Local Petrov-Galerkin (MLPG) “Mixed” approach. In this MLPG mixed approach, both the strains as well as displacements are interpolated, at randomly distributed points in the domain, through local meshless interpolation schemes such as the moving least squares(MLS) or radial basis fun...

In this paper, an H-Galerkin mixed finite element method is discussed for the coupled Burgers equations. The optimal error estimates of the semi-discrete and fully discrete schemes of the coupled Burgers equation are derived. Keywords—The coupled Burgers equation; H-Galerkin mixed finite element method; Backward Euler’s method; Optimal error estimates.

We investigate a periodic, one-dimensional, linear, and degenerate advection-diffusion equation. The problem is hyperbolic in a subinterval and parabolic in the complement, and the boundary conditions only impose the periodicity of the advective-diffusive flux to ensure mass conservation. Following Gastaldi and Quarteroni (1989), a condition is added to select the “physically acceptable” soluti...

The central scheme of Nessyahu and Tadmor [J. Comput. Phys., 87 (1990), pp. 408–463] solves hyperbolic conservation laws on a staggered mesh and avoids solving Riemann problems across cell boundaries. To overcome the difficulty of excessive numerical dissipation for small time steps, the recent work of Kurganov and Tadmor [J. Comput. Phys., 160 (2000), pp. 241–282] employs a variable control vo...

In this paper we generalize a new type of limiters based on the weighted essentially nonoscillatory (WENO) finite volume methodology for the Runge-Kutta discontinuous Galerkin (RKDG) methods solving nonlinear hyperbolic conservation laws, which were recently developed in [31] for structured meshes, to two-dimensional unstructured triangular meshes. The key idea of such limiters is to use the en...

In an earlier study (Zhang & Shu 2010b J. Comput. Phys. 229, 3091–3120 (doi:10.1016/ j.jcp.2009.12.030), genuinely high-order accurate finite volume and discontinuous Galerkin schemes satisfying a strict maximum principle for scalar conservation laws were developed. The main advantages of such schemes are their provable high-order accuracy and their easiness for generalization to multi-dimensio...