Jing-Mei Qiu

Positivity preserving semi-Lagrangian discontinuous Galerkin formulation: Theoretical analysis and application to the Vlasov-Poisson system

Semi-Lagrangian (SL) methods have been very popular in the Vlasov simulation community [29,3,4,18,32,5,24,26]. 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 [6], as a way to decouple the nonlinear Vlasov system into a sequence of 1-D advection equations, each of which has an advection velocity that only depends on coordinates that are transverse to the direction of propagation. To evolve the decoupled linear equations, we propose to couple the SL framework with the semi-discrete DG formulation. The proposed SL DG method is free of time step restriction compared with the Runge-Kutta DG method, which is known to suffer from numerical time step limitation with relatively small CFL numbers according to linear stability analysis. We apply the recently developed positivity preserving (PP) limiter [37], which is a low-cost black box procedure, to our scheme to ensure the positivity of the unknown probability density function without affecting the high order accuracy of the base SL DG scheme. We analyze the stability and accuracy properties of the SL DG scheme by establishing its connection with the direct and weak formulations of the characteristics/Lagrangian Galerkin method [23]. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and via classical plasma problems, such as Landau damping and the two stream instability.

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.

Conservative high order semi-Lagrangian finite difference WENO methods for advection in incompressible flow

In this paper, we propose a semi-Lagrangian finite difference formulation for approximating conservative form of advection equations with general variable coefficients. Compared with the traditional semi-Lagrangian finite difference schemes 5,25], which approximate the advective form of the equation via direct characteristics tracing, the scheme proposed in this paper approximates the conservative form of the equation. This essential difference makes the proposed scheme naturally conservative for equations with general variable coefficients. The proposed conservative semi-Lagrangian finite difference framework is coupled with high order essentially non-oscillatory (ENO) or weighted ENO (WENO) reconstructions to achieve high order accuracy in smooth parts of the solution and to capture sharp interfaces without introducing spurious oscillations. The scheme is extended to high dimensional problems by Strang splitting. The performance of the proposed schemes is demonstrated by linear advection, rigid body rotation, swirling deformation, and two dimensional incompressible flow simulation in the vorticity stream-function formulation. As the information is propagating along characteristics, the proposed scheme does not have the CFL time step restriction of the Eulerian method, allowing for a more efficient numerical realization for many application problems.

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

In this paper, we propose a finite difference AMR-WENO method for hyperbolic conservation laws. The proposed method combines the adaptive mesh refinement (AMR) framework [4,5] with the high order finite difference weighted essentially non-oscillatory (WENO) method in space and the total variation diminishing (TVD) Runge-Kutta (RK) method in time (WENO-RK) [18,10] by a high order coupling. Our goal is to realize mesh adaptivity in the AMR framework, while maintaining very high (higher than second) order accuracy of the WENO-RK method in the finite difference setting. The high order coupling of AMR and WENO-RK is accomplished by high order prolongation in both space (WENO interpolation) and time (Hermite interpolation) from coarse to fine grid solutions, and at ghost points. The resulting AMR-WENO method is accurate, robust and efficient, due to the mesh adaptivity and very high order spatial and temporal accuracy. We have experimented with both the third and the fifth order AMR-WENO schemes. We demonstrate the accuracy of the proposed scheme using smooth test problems, and their quality and efficiency using several 1D and 2D nonlinear hyperbolic problems with very challenging initial conditions. The AMR solutions are observed to perform as well as, and in some cases even better than, the corresponding uniform fine grid solutions. We conclude that there is significant improvement of the fifth order AMR-WENO over the third order one, not only in accuracy for smooth problems, but also in its ability in resolving complicated solution structures, due to the very low numerical diffusion of high order schemes. In our work, we found that it is difficult to design a robust AMR-WENO scheme that is both conservative and high order (higher than second order), due to the mass inconsistency of coarse and fine grid solutions at the initial stage in a finite difference scheme. Resolving these issues as well as conducting comprehensive evaluation of computational efficiency constitute our future work.

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

Spectral deferred correction (SDC) methods for solving ordinary differential equations (ODEs) were introduced by Dutt, Greengard and Rokhlin (2000). It was shown in that paper that SDC methods can achieve arbitrary high order accuracy and possess nice stability properties. Their SDC methods are constructed with low order integrators, such as forward Euler or backward Euler, and are able to handle stiff and non-stiff terms in the ODEs. In this paper, we use high order Runge-Kutta (RK) integrators to construct a family of related methods, which we refer to as integral deferred correction (IDC) methods. The distribution of quadrature nodes is assumed to be uniform, and the corresponding local error analysis is given. The smoothness of the error vector associated with an IDC method, measured by the discrete Sobolev norm, is a crucial tool in our analysis. The expected order of accuracy is demonstrated through several numerical examples. Superior numerical stability and accuracy regions are observed when high order RK integrators are used to construct IDC methods.

A Conservative Semi-Lagrangian Discontinuous Galerkin Scheme on the Cubed Sphere

AbstractThe discontinuous Galerkin (DG) methods designed for hyperbolic problems arising from a wide range of applications are known to enjoy many computational advantages. DG methods coupled with strong-stability-preserving explicit Runge–Kutta discontinuous Galerkin (RKDG) time discretizations provide a robust numerical approach suitable for geoscience applications including atmospheric modeling. However, a major drawback of the RKDG method is its stringent Courant–Friedrichs–Lewy (CFL) stability restriction associated with explicit time stepping. To address this issue, the authors adopt a dimension-splitting approach where a semi-Lagrangian (SL) time-stepping strategy is combined with the DG method. The resulting SLDG scheme employs a sequence of 1D operations for solving multidimensional transport equations. The SLDG scheme is inherently conservative and has the option to incorporate a local positivity-preserving filter for tracers. A novel feature of the SLDG algorithm is that it can be used for mult...

Parametrized Positivity Preserving Flux Limiters for the High Order Finite Difference WENO Scheme Solving Compressible Euler Equations

In this paper, we develop parametrized positivity satisfying flux limiters for the high order finite difference Runge–Kutta weighted essentially non-oscillatory scheme solving compressible Euler equations to maintain positive density and pressure. Negative density and pressure, which often leads to simulation blow-ups or nonphysical solutions, emerges from many high resolution computations in some extreme cases. The methodology we propose in this paper is a nontrivial generalization of the parametrized maximum principle preserving flux limiters for high order finite difference schemes solving scalar hyperbolic conservation laws (Liang and Xu in J Sci Comput 58:41–60, 2014; Xiong et al. in J Comput Phys 252:310–331, 2013; Xu in Math Comput 83:2213–2238, 2014). To preserve the maximum principle, the high order flux is limited towards a first order monotone flux, where the limiting procedures are designed by decoupling linear maximum principle constraints. High order schemes with such flux limiters are shown to preserve the high order accuracy via local truncation error analysis and by extensive numerical experiments with mild CFL constraints. The parametrized flux limiting approach is generalized to the Euler system to preserve the positivity of density and pressure of numerical solutions via decoupling some nonlinear constraints. Compared with existing high order positivity preserving approaches (Zhang and Shu in Proc R Soc A Math Phys Eng Sci 467:2752–2776, 2011; J Comput Phys 230:1238–1248, 2011; J Comput Phys 231:2245–2258, 2012), our proposed algorithm is positivity preserving by the design; it is computationally efficient and maintains high order spatial and temporal accuracy in our extensive numerical tests. Numerical tests are performed to demonstrate the efficiency and effectiveness of the proposed new algorithm.

A parametrized maximum principle preserving flux limiter for finite difference RK-WENO schemes with applications in incompressible flows

In Xu (2013) [14], a class of parametrized flux limiters is developed for high order finite difference/volume essentially non-oscillatory (ENO) and Weighted ENO (WENO) schemes coupled with total variation diminishing (TVD) Runge-Kutta (RK) temporal integration for solving scalar hyperbolic conservation laws to achieve strict maximum principle preserving (MPP). In this paper, we continue along this line of research, but propose to apply the parametrized MPP flux limiter only to the final stage of any explicit RK method. Compared with the original work (Xu, 2013) [14], the proposed new approach has several advantages: First, the MPP property is preserved with high order accuracy without as much time step restriction; Second, the implementation of the parametrized flux limiters is significantly simplified. Analysis is performed to justify the maintenance of third order spatial/temporal accuracy when the MPP flux limiters are applied to third order finite difference schemes solving general nonlinear problems. We further apply the limiting procedure to the simulation of the incompressible flow: the numerical fluxes of a high order scheme are limited toward that of a first order MPP scheme which was discussed in Levy (2005) [3]. The MPP property is guaranteed, while designed high order of spatial and temporal accuracy for the incompressible flow computation is not affected via extensive numerical experiments. The efficiency and effectiveness of the proposed scheme are demonstrated via several test examples.

A high order time splitting method based on integral deferred correction for semi-Lagrangian Vlasov simulations

Semi-Lagrangian schemes with various splitting methods, and with different reconstruction/interpolation strategies have been applied to kinetic simulations. For example, the order of spatial accuracy of the algorithms proposed in Qiu and Christlieb (2010) [29] is very high (as high as ninth order). However, the temporal error is dominated by the operator splitting error, which is second order for Strang splitting. It is therefore important to overcome such low order splitting error, in order to have numerical algorithms that achieve higher orders of accuracy in both space and time. In this paper, we propose to use the integral deferred correction (IDC) method to reduce the splitting error. Specifically, the temporal order accuracy is increased by r with each correction loop in the IDC framework, where r=1,2r=1,2 for coupling the first order splitting and the Strang splitting, respectively. The proposed algorithm is applied to the Vlasov–Poisson system, the guiding center model, and two dimensional incompressible flow simulations in the vorticity stream-function formulation. We show numerically that the IDC procedure can automatically increase the order of accuracy in time. We also investigate numerical stability of the proposed algorithm via performing Fourier analysis to a linear model problem.

Hybrid semi-Lagrangian finite element-finite difference methods for the Vlasov equation

In this paper, we propose a new conservative hybrid finite element-finite difference method for the Vlasov equation. The proposed methodology uses Strang splitting to decouple the nonlinear high dimensional Vlasov equation into two lower dimensional equations, which describe spatial advection and velocity acceleration/deceleration processes respectively. We then propose to use a semi-Lagrangian (SL) discontinuous Galerkin (DG) scheme (or Eulerian Runge-Kutta (RK) DG scheme with local time stepping) for spatial advection, and use a SL finite difference WENO for velocity acceleration/deceleration. Such hybrid method takes the advantage of DG scheme in its compactness and its ability in handling complicated spatial geometry; while takes the advantage of the WENO scheme in its robustness in resolving filamentation solution structures of the Vlasov equation. The proposed highly accurate methodology enjoys great computational efficiency, as it allows one to use relatively coarse phase space mesh due to the high order nature of the scheme. At the same time, the time step can be taken to be extra large in the SL framework. The quality of the proposed method is demonstrated via basic test problems, such as linear advection and rigid body rotation, and classical plasma problems, such as Landau damping and the two stream instability. Although we only tested 1D1V examples, the proposed method has the potential to be extended to problems with high spatial dimensions and complicated geometry. This constitutes our future research work.