Andrew Christlieb

A conservative high order semi-Lagrangian WENO method for the Vlasov equation

In this paper, we propose a novel Vlasov solver based on a semi-Lagrangian method which combines Strang splitting in time with high order WENO (weighted essentially non-oscillatory) reconstruction in space. A key insight in this work is that the spatial interpolation matrices, used in the reconstruction process of a semi-Lagrangian approach to linear hyperbolic equations, can be factored into right and left flux matrices. It is the factoring of the interpolation matrices which makes it possible to apply the WENO methodology in the reconstruction used in the semi-Lagrangian update. The spatial WENO reconstruction developed for this method is conservative and updates point values of the solution. While the third, fifth, seventh and ninth order reconstructions are presented in this paper, the scheme can be extended to arbitrarily high order. WENO reconstruction is able to achieve high order accuracy in smooth parts of the solution while being able to capture sharp interfaces without introducing oscillations. Moreover, the CFL time step restriction of a regular finite difference or finite volume WENO scheme is removed in a semi-Lagrangian framework, allowing for a cheaper and more flexible numerical realization. The quality of the proposed method is demonstrated by applying the approach to basic test problems, such as linear advection and rigid body rotation, and to classical plasma problems, such as Landau damping and the two-stream instability. Even though the method is only second order accurate in time, our numerical results suggest the use of high order reconstruction is advantageous when considering the Vlasov-Poisson system.

Parallel High-Order Integrators

In this work we discuss a class of defect correction methods which is easily adapted to create parallel time integrators for multicore architectures and is ideally suited for developing methods which can be order adaptive in time. The method is based on integral deferred correction (IDC), which was itself motivated by spectral deferred correction by Dutt, Greengard, and Rokhlin [BIT, 40 (2000), pp. 241-266]. The method presented here is a revised formulation of explicit IDC, dubbed revisionist IDC (RIDC), which can achieve

Adaptive mesh refinement based on high order finite difference WENO scheme for multi-scale simulations

p

Integral deferred correction methods constructed with high order Runge-Kutta integrators

th-order accuracy in “wall-clock time” comparable to a single forward Euler simulation on problems where the time to evaluate the right-hand side of a system of differential equations is greater than latency costs of interprocessor communication, such as in the case of the

Adaptive sub-linear time Fourier algorithms

N

Grid-free plasma Simulation techniques

-body problem. The key idea is to rewrite the defect correction framework so that, after initial start-up costs, each correction loop can be lagged behind the previous correction loop in a manner that facilitates running the predictor and

High accuracy solutions to energy gradient flows from material science models

M=p-1

Energy-conserving discontinuous Galerkin methods for the Vlasov-Ampère system

correctors in parallel on an interval which has

Gridless DSMC

K

Efficient particle Simulation of a virtual cathode using a grid-free treecode Poisson solver

steps, where