An adaptive ADER finite volume method on unstructured meshes is proposed. The method combines high order polyharmonic spline WENO reconstruction with high order flux evaluation. Polyharmonic splines are utilised in the recovery step of the finite volume method yielding a WENO reconstruction that is stable, flexible and optimal in the associated Sobolev (BeppoLevi) space. The flux evaluation is ...

Abstract In this article, we briefly review the random choice method (RCM) and ADER methods for solving one two-dimensional hyperbolic conservation laws. The main advantage of RCM is that it computes discontinuities with infinite resolution. method, original problem reduced to a set local Riemann problems (RPs). exact solutions these RPs are used form solution problem. However, has following di...

We present a theoretical solution for the Riemann problem for the five-equation two-phase non-conservative model of Saurel and Abgrall. This solution is then utilized in the construction of upwind non-conservative methods to solve the general initial-boundary value problem for the two-phase flow model in non-conservative form. The basic upwind scheme constructed is the non-conservative analogue...

A conservative 3-D discontinuous Galerkin (DG) baroclinic model has been developed in the NCAR High-Order Method Modeling Environment (HOMME) to investigate global atmospheric flows. The computational domain is a cubed-sphere free from coordinate singularities. The DG discretization uses a high-order nodal basis set of orthogonal Lagrange-Legendre polynomials and fluxes of inter-element boundar...

A class of high order nodal discontinuous Galerkin implicit-explicit (DG-IMEX) schemes with asymptotic preserving (AP) property has been developed for the one-dimensional (1D) BGK equation in Xiong et. al. (2015) [40], based on a micro-macro reformulation. The schemes are globally stiffly accurate and asymptotically consistent, and as the Knudsen number becomes small or goes to zero, they recov...

Abstract This paper is on arbitrary high order fully discrete one-step ADER discontinuous Galerkin schemes with subcell finite volume limiters applied to a new class of first hyperbolic reformulations nonlinear dispersive systems based an extended Lagrangian approach introduced by Dhaouadi et al. (Stud Appl Math 207:1–20, 2018), Favrie and Gavrilyuk (Nonlinearity 30:2718–2736, 2017). We conside...

Low voltage microgrids including sensitive loads often face unbalanced load conditions. Therefore, a compensation procedure should be carried out in order to balance and restore sensitive load’s voltage. In this paper, an effective voltage control strategy has been proposed for the autonomous operation of microgrids, under unbalanced load conditions.  The proposed strategy balances single-phase...

In this paper, a general high-order multi-domain hybrid DG/WENO-FDmethod, which couples a p-order (p ≥ 3) DG method and a q-order (q ≥ 3) WENO-FD scheme, is developed. There are two possible coupling approaches at the domain interface, one is non-conservative, the other is conservative. The non-conservative coupling approach can preserve optimal order of accuracy and the local conservative erro...

Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community [28, 2, 3, 18, 31, 4, 23, 25]. In this paper, we propose a new Strang split SL discontinuous Galerkin (DG) method for solving the Vlasov equation. Specifically, we apply the Strang splitting for the Vlasov equation [5], as a way to decouple the nonlinear Vlasov system into a sequence of linear equations. To e...