A WENO-based Method of Lines Transpose approach for Vlasov simulations

Abstract

In this paper, a high order implicit Method of Line Transpose (MOL T ) method based on a weighted essentially non-oscillatory (WENO) methodology is developed for one-dimensional linear transport equations and further applied to the Vlasov-Poisson (VP) simulations via dimensional splitting. In the MOL T framework, the time variable is first discretized by a diagonally implicit strong-stability-preserving Runge-Kutta method, resulting in a boundary value problem (BVP) at the discrete time levels. Then an integral formulation coupled with a high order WENO methodology is employed to solve the BVP. As a result, the proposed scheme is high order accurate in both space and time and free of oscillations even though the solution is discontinuous or has sharp gradients. Moreover, the scheme is able to take larger time step evolution compared with an explicit MOL WENO scheme with the same order of accuracy. The desired positivity-preserving (PP) property of the scheme is further attained by incorporating a newly proposed high order PP limiter. We perform numerical experiments on several benchmarks including linear advection, solid body rotation problem; and on the Landau damping, two-stream instabilities, bump-on-tail, and plasma sheath by solving the VP system. The efficacy and efficiency of the proposed scheme is numerically verified.

High-resolution shock-capturing numerical simulations of three-phase immiscible fluids from the unsaturated to the saturated zone

“…Hybrid combinations are also possible 9 . Other used methods are: transversal methods of lines for the numerical modeling of vertical infiltration into the vadose zone 28 , 29 , or a WENO based method of lines 30 . Although there were very few attempts to use a supercomputer to simulate subsurface solute transport, the situation has changed in the last ten years.. New codes use HPC, for example, PFLOTRAN 31 , 32 , Parflow Hydrologic model 33 , 34 , RichardsFOAM 35 , CATHY (CATchment HYdrology) 36 .…”

A Kernel Based Unconditionally Stable Scheme for Nonlinear Parabolic Partial Differential Equations

“…Over the past several years, the MOL T methods have been developed for solving the heat equation [4,6,20,24], Maxwell's equations [8], the advection equation and Vlasov equation [9], among others. This methodology can be generalized to solving some nonlinear problems, such as the Cahn-Hilliard equation [4].…”

A hybrid HWENO-based method of lines transpose approach for Vlasov simulations

“…Recently, an alternative approach to advance the solution called the method of line transpose (MOL T ) is well designed to solve the VP system [23] . This method is also known as the Rothe's method or transversal method of lines in the literature [24,25] .…”

“…Moreover, this method is capable of conveniently handling general boundary conditions. This MOL T approach has been applied to solve the heat equation, the Allen-Cahn equation [26] , Maxwell's equations [27] , transport equation and the VP system [23] . Note that the MOL T framework is rarely applied to general nonlinear problems, mainly because efficient algorithms of inverting nonlinear BVPs are lacking.…”