Conservative and non-conservative methods based on hermite weighted essentially-non-oscillatory reconstruction for Vlasov equations

We introduce a WENO reconstruction based on Hermite interpolation both for semi-Lagrangian and finite difference methods. This WENO reconstruction technique allows to control spurious oscillations. We develop third and fifth order methods and apply them to non-conservative semi-Lagrangian schemes and conservative finite difference methods. Our numerical results will be compared to the usual semi-Lagrangian method with cubic spline reconstruction and the classical fifth order WENO finite difference scheme. These reconstructions are observed to be less dissipative than the usual weighted essentially non- oscillatory procedure. We apply these methods to transport equations in the context of plasma physics and the numerical simulation of turbulence phenomena

Mixed semi-Lagrangian/finite difference methods for plasma simulations

In this paper, we present an efficient algorithm for the long time behavior of plasma simulations. We will focus on 4D drift-kinetic model, where the plasma's motion occurs in the plane perpendicular to the magnetic field and can be governed by the 2D guiding-center model. Hermite WENO reconstructions, already proposed in \cite{YF15}, are applied for solving the Vlasov equation. Here we consider an arbitrary computational domain with an appropriate numerical method for the treatment of boundary conditions. Then we apply this algorithm for plasma turbulence simulations. We first solve the 2D guiding-center model in a D-shape domain and investigate the numerical stability of the steady state. Then, the 4D drift-kinetic model is studied with a mixed method, i.e. the semi-Lagrangian method in linear phase and finite difference method during the nonlinear phase. Numerical results show that the mixed method is efficient and accurate in linear phase and it is much stable during the nonlinear phase. Moreover, in practice it has better conservation properties.Comment: arXiv admin note: text overlap with arXiv:1312.448

Higher-Order Finite Difference Scheme With TVD Filter Algorithm. Part 1: Simulations of Scalar Advective Dominant Problems.

Abstract Engquist et al. (1989) proposed non-linear TVD filters. When combined with a traditional higher-order finite difference scheme, these filters can simulate shock or high concentration gradient problems with no spurious oscillations. Among the filter proposed is the filter algorithm 2.2. However, this algorithm flattens extrema that are not the results of overshooting and consequency the scheme reduces to a low order of accuracy locally around smooth extrema. Modification of the TVD filter algorithm 2.2 has been proposed in this paper to overcome this problem. Several conservative finite difference schemes are considered for testing the TVD filter.  Non-conservative schemes consisting of 4th and 6th -order Runge-Kutta method are also evaluated. The modified filter has been tested to simulate seven test cases, including a pure advection of scalar profiles, a pure advection with variable velocity, two inviscid burger equations, an advection-diffusion equation with variable velocity and dispersion, advection of three solid bodies in rotating fluid around a square of a side length of 2, and a two-dimensional advection-diffusion equation. The numerical experiments showed that applying the modified TVD filter, combined with the higher-order non-TVD finite difference schemes for solving the advection equation, can produce accurate solutions with no oscillations and no clipping effect extrema. Keywords : Non-Oscillatory schemes, TVD filter, advection, higher-order accurate Abstrak Engquist dkk. (1989) mengusulkan filter TVD non-linear. Ketika dikombinasikan dengan skema beda hingga orde tinggi tradisional, filter ini dapat mensimulasikan kejutan atau masalah gradien konsentrasi tinggi tanpa osilasi palsu. Di antara filter yang diusulkan adalah algoritma filter 2.2. Namun, algoritma ini meratakan ekstrem yang bukan merupakan hasil dari “overshooting” dan konsekuensinya skema tersebut direduksi ke tingkat akurasi yang rendah secara lokal di sekitar ekstrem halus. Modifikasi algoritma filter TVD 2.2 telah diusulkan dalam makalah ini untuk mengatasi masalah ini. Beberapa skema konservatif dipertimbangkan untuk menguji filter TVD. Skema non-konservatif metode Runge-Kutta orde ke-4 dan ke-6 juga dievaluasi. Filter yang dimodifikasi diuji untuk mensimulasikan tujuh kasus, termasuk adveksi murni profil skalar, adveksi murni dengan kecepatan bervariasi, dua persamaan burger inviscid, persamaan adveksi-difusi dengan kecepatan bervariasi dan dispersi, adveksi tiga benda padat berputar oleh aliran di bidang persegi dengan panjang sisi 2, dan persamaan adveksi-difusi dua dimensi. Eksperimen numerik menunjukkan bahwa penerapan filter TVD yang dimodifikasi, dikombinasikan dengan skema beda hingga non-TVD orde tinggi untuk menyelesaikan persamaan adveksi, dapat menghasilkan solusi yang akurat tanpa osilasi dan tanpa efek kliping Kata Kunci : skema tanpa osilasi, Filter TVD, advection, orde-akurasi tingg