Abstract. -A new splitting is proposed for solving the Vlasov-Maxwell system. This splitting is based on a decomposition of the Hamiltonian of the Vlasov-Maxwell system and allows for the construction of arbitrary high order methods by composition (independent of the specific deterministic method used for the discretization of the phase space). Moreover, we show that for a spectral method in space this scheme satisfies Poisson's equation without explicitly solving it. Finally, we present some examples in the context of the time evolution of an electromagnetic plasma instability which emphasizes the excellent behavior of the new splitting compared to methods from the literature.
Many problems encountered in plasma physics require a description by kinetic equations, which are posed in an up to six-dimensional phase space. A direct discretization of this phase space, often called the Eulerian approach, has many advantages but is extremely expensive from a computational point of view. In the present paper we propose a dynamical lowrank approximation to the Vlasov-Poisson equation, with time integration by a particular splitting method. This approximation is derived by constraining the dynamics to a manifold of low-rank functions via a tangent space projection and by splitting this projection into the subprojections from which it is built. This reduces a time step for the six-(or four-) dimensional Vlasov-Poisson equation to solving two systems of three-(or two-) dimensional advection equations over the time step, once in the position variables and once in the velocity variables, where the size of each system of advection equations is equal to the chosen rank. By a hierarchical dynamical low-rank approximation, a time step for the Vlasov-Poisson equation can be further reduced to a set of six (or four) systems of one-dimensional advection equations, where the size of each system of advection equations is still equal to the rank. The resulting systems of advection equations can then be solved by standard techniques such as semi-Lagrangian or spectral methods. Numerical simulations in two and four dimensions for linear Landau damping, for a two-stream instability and for a plasma echo problem highlight the favorable behavior of this numerical method and show that the proposed algorithm is able to drastically reduce the required computational effort.the numerical algorithms used (see, for example, [9,15,13,19,23,30,46,49,50,53,3,7,52,5,6]) as well as the corresponding parallelization to state of the art HPC systems (see, for example, [51,12,2,35,42,11,8]).The seminal paper [3] introduced a time-splitting approach. This approach has the advantage that the nonlinear Vlasov-Poisson equation can be reduced to a sequence of one-dimensional advection equations (see, for example, [18,20,14]). Splitting schemes that have similar properties have been proposed for a range of more complicated models (including the relativistic Vlasov-Maxwell equations [52,5], drift-kinetic models [23,7], and gyrokinetic models [22]). This is then usually combined with a semi-Lagrangian discretization of the resulting advection equations. A semi-Lagrangian approach has the advantage that the resulting numerical method is completely free from a Courant-Friedrichs-Lewy (CFL) condition.Nevertheless, all these methods still incur a computational and storage cost that scales as O(n 2d ). Thus, it seems natural to ask whether any of the dimension reduction techniques developed for high-dimensional problems are able to help alleviate the computational burden of the Eulerian approach. To understand why it is not entirely clear if such an approach could succeed, we will consider the following equation (withThis is just the one-dime...
For diffusion-reaction equations employing a splitting procedure is attractive as it reduces the computational demand and facilitates a parallel implementation. Moreover, it opens up the possibility to construct second-order integrators that preserve positivity independent of the time step size used. However, for boundary conditions that are neither periodic nor of homogeneous Dirichlet type, order reduction limits its usefulness. In the situation described the Strang splitting procedure is not more accurate than Lie splitting. In this paper, we propose a modified Lie/Strang splitting procedure that, while retaining all the favorable properties of the original method, does not suffer from order reduction. That is, the modified Strang splitting is second-order accurate in time. We demonstrate our results by conducting numerical simulations in one and two space dimensions with inhomogeneous and time-dependent Dirichlet boundary conditions. In addition, a mathematically rigorous convergence analysis is conducted that confirms the results observed in the numerical simulations.
Abstract. Splitting methods constitute a well-established class of numerical schemes for the time integration of partial differential equations. Their main advantages over more traditional schemes are computational efficiency and superior geometric properties. In the presence of non-trivial boundary conditions, however, splitting methods usually suffer from order reduction and some additional loss of accuracy. For diffusion-reaction equations with inhomogeneous oblique boundary conditions, a modification of the classic second-order Strang splitting is proposed that successfully resolves the problem of order reduction. The same correction also improves the accuracy of the classic first-order Lie splitting. The proposed modification only depends on the available boundary data and, in the case of non Dirichlet boundary conditions, on the computed numerical solution. Consequently, this modification can be implemented in an efficient way, which makes the modified splitting schemes superior to their classic versions. The framework employed in our error analysis also allows us to explain the fractional orders of convergence that are often encountered for classic Strang splitting. Numerical experiments that illustrate the theory are provided.Key words. Strang splitting, diffusion-reaction equation, oblique boundary conditions, Neumann boundary conditions, Robin boundary conditions, mixed boundary conditions, order reduction AMS subject classifications. 65M20, 65M12, 65L041. Introduction. For the numerical solution of evolution equations, splitting methods have attracted much attention in recent years. The main advantage of splitting methods over more traditional time integration schemes is the fact that the partial vector fields in the splitting can usually be integrated more efficiently, sometimes much more efficiently, compared to applying a numerical method to the full problem. Moreover, splitting methods have, in general, better geometric properties [10]. As a typical example, we mention splitting of the Schrödinger equation [6] into the kinetic and the potential part. The former is integrated in frequency space using the fast Fourier transform, whereas the latter can often be integrated exactly. This requires considerably less computational effort compared to directly applying, for example, an implicit Runge-Kutta method to the Schrödinger equation.In this paper, we will consider a splitting approach for diffusion-reaction equations. In our problem, the diffusion is modelled by a linear elliptic differential operator and the reaction by a nonlinear smooth function. Such a splitting into a linear elliptic problem, which can be solved efficiently by fast Poisson techniques [17], and a nonlinear ordinary differential equation, which is simply solved pointwise in space, has attracted much attention in the literature (see, e.g., the articles [1,3,5,8,11] and the monograph [14]).Although splitting methods are widely used in multiphysics problems, a rigorous error analysis is still missing for many important applications....
Abstract.A rigorous convergence analysis of the Strang splitting algorithm for Vlasov-type equations in the setting of abstract evolution equations is provided. It is shown that under suitable assumptions the convergence is of second order in the time step τ . As an example, it is verified that the Vlasov-Poisson equations in 1+1 dimensions fit into the framework of this analysis. Further, numerical experiments for the latter case are presented.
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