Search

In this paper, we apply semi-Lagrangian discontinuous Galerkin (SLDG) methods for linear hyperbolic equations in one space dimension and analyze the error between the numerical and exact solutions under the L2-norm. In all the previous works, the theoretical analysis of the SLDG method would suggest a suboptimal convergence rate due to the error accumulation over time steps. However, numerical experiments demonstrate an optimal convergence rate and, if the terminal time is large, a superconvergence rate. In this paper, we will prove optimal convergence and optimal superconvergence rates. There are three main difficulties: 1. The error analysis on overlapping meshes. Due to the nature of the semi-Lagrangian time discretization, we need to introduce the background Eulerian mesh and the shifted mesh. The two meshes are staggered, and it is not easy to construct local projections and to handle the error accumulation during time evolution. 2. The superconvergence of time-dependent terms under the L2-norm. The error of the numerical and exact solutions can be divided into two parts, the projection error and the time-dependent superconvergence term. The projection strongly depends on the superconvergence rates. Therefore, we need to construct a sequence of projections and gradually improve the superconvergence rates. 3. The stopping criterion of the sequence of projections. The sequence of projections are basically of the same form. We need to show that the projections exist up to some certain order since the superconvergence rate cannot be infinity. Hence, we will seek some ''hidden'' condition for the existence of the projections. In this paper, we will solve all the three difficulties and construct several local projections to prove the optimal convergence and superconvergence rates. Numerical experiments verify the theoretical findings. [ABSTRACT FROM AUTHOR]

We propose a new Eulerian-Lagrangian Runge-Kutta finite volume method for numerically solving convection and convection-diffusion equations. Eulerian-Lagrangian and semi-Lagrangian methods have grown in popularity mostly due to their ability to allow large time steps. Our proposed scheme is formulated by integrating the PDE on a space-time region partitioned by approximations of the characteristics determined from the Rankine-Hugoniot jump condition; and then rewriting the time-integral form into a time differential form to allow application of Runge-Kutta (RK) methods via the method-of-lines approach. The scheme can be viewed as a generalization of the standard Runge-Kutta finite volume (RK-FV) scheme for which the space-time region is partitioned by approximate characteristics with zero velocity. The high-order spatial reconstruction is achieved using the recently developed weighted essentially non-oscillatory schemes with adaptive order (WENO-AO); and the high-order temporal accuracy is achieved by explicit RK methods for convection equations and implicit-explicit (IMEX) RK methods for convection-diffusion equations. Our algorithm extends to higher dimensions via dimensional splitting. Numerical experiments demonstrate our algorithm's robustness, high-order accuracy, and ability to handle extra large time steps. • The novel design of the EL RK FV framework that accommodates both the SL FV method and classical Eulerian RK FV method. • Partition of space-time region that follows dynamics of characteristics. • The time step constraint from an Eulerian method is greatly mitigated. [ABSTRACT FROM AUTHOR]

In this paper, we present a fourth-order conservative semi-Lagrangian (SL) finite volume (FV) weighted essentially non-oscillatory (WENO) scheme without operator splitting for two-dimensional linear transport equations with applications in kinetic models including the nonlinear Vlasov–Poisson system, the guiding center Vlasov model and the incompressible Euler equation in the vorticity-stream function formulation. To achieve fourth-order accuracy in space, two main ingredients are proposed in the SL FV formulation. Firstly, we introduce a so-called cubic-curved quadrilateral upstream cell and applying an efficient clipping method to evaluate integrals on upstream cells. Secondly, we construct a new WENO reconstruction operator, which recovers a P 3 polynomial from neighboring cell averages. Mass conservation is accomplished with the mass conservative nature of the reconstruction operator and the SL formulation. A positivity-preserving limiter is applied to maintain the positivity of the numerical solution wherever appropriate. For nonlinear kinetic models, the SL scheme is coupled with a fourth-order Runge–Kutta exponential integrator for high-order temporal accuracy. Extensive benchmarks are tested to verify the designed properties. [ABSTRACT FROM AUTHOR]

Transport equations arise in many applications such as rarefied gas dynamics, neutron transport, and radiative transfer. In this work, we consider some linear kinetic transport equations in a diffusive scaling and design high order asymptotic preserving (AP) methods within the discontinuous Galerkin method framework, with the main objective to achieve unconditional stability in the diffusive regime when the Knudsen number ε ≪ 1 , and to achieve high order accuracy when ε = O (1) and when ε ≪ 1. Initial layers are also taken into account. The ingredients to accomplish our goal include: model reformulations based on the micro-macro decomposition and the limiting diffusive equation, local discontinuous Galerkin (LDG) methods in space, globally stiffly accurate implicit-explicit (IMEX) Runge-Kutta methods in time, and strategies to handle non-well prepared initial data. Formal asymptotic analysis is carried out for the continuous model within the micro-macro decomposed framework to derive the initial layer as well as the interior problem with an asymptotically consistent initial condition as ε → 0 , and it is also conducted for numerical schemes to show the AP property and to understand the numerical initial treatments in the presence of initial layers. Fourier type stability analysis is performed, and it confirms the unconditional stability in the diffusive regime, and moreover it gives the stability condition in the kinetic regime when ε = O (1). In the reformulation step, a weighted diffusive term is added and subtracted to remove the parabolic stiffness and enhance the numerical stability in the diffusive regime. Such idea is not new, yet our numerical stability and asymptotic analysis provide new mathematical understanding towards the desired properties of the weight function. Finally, numerical examples are presented to demonstrate the accuracy, stability, and asymptotic preserving property of the proposed methods, as well as the effectiveness of the proposed strategies in the presence of the initial layer. • High order AP methods within the DG framework for kinetic transport models. • Fourier-type stability analysis for kinetic and diffusive regimes. • Unconditional stability in the diffusive regime. • AP property in the presence of initial layers. • Advancement in mathematical understanding of some weight function. [ABSTRACT FROM AUTHOR]

• The uniformly high order accuracy (e.g. third order) in space and in time. • Local mass conservation. • Stability under large time stepping size and unconditional stability for linear problems. • Compactness. In this paper, we propose an efficient high order semi-Lagrangian (SL) discontinuous Galerkin (DG) method for solving linear convection-diffusion equations. The method generalizes our previous work on developing the SLDG method for transport equations [5] , making it capable of handling additional diffusion and source terms. Within the DG framework, the solution is evolved along the characteristics; while the diffusion term is discretized by the local DG (LDG) method and integrated along characteristics by implicit Runge-Kutta methods together with source terms. The proposed method is named the 'SLDG-LDG' method and enjoys many attractive features of the DG and SL methods. These include the uniformly high order accuracy (e.g. third order) in space and in time, compact, mass conservative, and stability under large time stepping size. An L 2 stability analysis is provided when the method is coupled with the first order backward Euler discretization. Effectiveness of the method are demonstrated by a group of numerical tests in one and two dimensions. [ABSTRACT FROM AUTHOR]