Charge-conserving grid based methods for the Vlasov–Maxwell equations

In this article, we introduce numerical schemes for the Vlasov–Maxwell equations relying on different kinds of grid-based Vlasov solvers, as opposite to PIC schemes, which enforce a discrete continuity equation. The idea underlying these schemes relies on a time-splitting scheme between configuration space and velocity space for the Vlasov equation and on the computation of the discrete current in a form that is compatible with the discrete Maxwell solver.     

A positivity-preserving high-order semi-Lagrangian discontinuous Galerkin scheme for the Vlasov–Poisson equations

The Vlasov–Poisson equations describe the evolution of a collisionless plasma, represented through a probability density function (PDF) that self-interacts via an electrostatic force. One of the main difficulties in numerically solving this system is the severe time-step restriction that arises from parts of the PDF associated with moderate-to-large velocities. The dominant approach in the plasma physics community for removing these time-step restrictions is the so-called particle-in-cell (PIC) method, which discretizes the distribution function into a set of macro-particles, while the electric field is represented on a mesh. Several alternatives to this approach exist, including fully Lagrangian, fully Eulerian, and so-called semi-Lagrangian methods. The focus of this work is the semi-Lagrangian approach, which begins with a grid-based Eulerian representation of both the PDF and the electric field, then evolves the PDF via Lagrangian dynamics, and finally projects this evolved field back onto the original Eulerian mesh. In particular, we develop in this work a method that discretizes the 1 + 1 Vlasov–Poisson system via a high-order discontinuous Galerkin (DG) method in phase space, and an operator split, semi-Lagrangian method in time. Second-order accuracy in time is relatively easy to achieve via Strang operator splitting. With additional work, using higher-order splitting and a higher-order method of characteristics, we also demonstrate how to push this scheme to fourth-order accuracy in time. We show how to resolve all of the Lagrangian dynamics in such a way that mass is exactly conserved, positivity is maintained, and high-order accuracy is achieved. The Poisson equation is solved to high-order via the smallest stencil local discontinuous Galerkin (LDG) approach. We test the proposed scheme on several standard test cases.     

保持总能量守恒的“半拉格朗日算法”

大气海洋问题数值计算的重要特征之一是需要作长时间的数值积分,因此在方程差分离散化后如何保持原问题的物理特性成为一个很关键的问题。从传统的"半拉格朗日法"和显式完全能量守恒差分法中吸取"营养",设计出一种既能保持总能量守恒又能增大时间步长的显式差分算法     

A Hamilton-Jacobi-Bellman approach to optimal trade execution

The optimal trade execution problem is formulated in terms of a mean-variance tradeoff, as seen at the initial time. The mean-variance problem can be embedded in a linear-quadratic (LQ) optimal stochastic control problem. A semi-Lagrangian scheme is used to solve the resulting nonlinear Hamilton-Jacobi-Bellman (HJB) PDE. This method is essentially independent of the form for the price impact functions. Provided a strong comparison property holds, we prove that the numerical scheme converges to the viscosity solution of the HJB PDE. Numerical examples are presented in terms of the efficient trading frontier and the trading strategy. The numerical results indicate that in some cases there are many different trading strategies which generate almost identical efficient frontiers.     

Developing new approaches for the path tubes method

The path tubes (PT) method for advection equations is a semi-Lagrangian algorithm recently introduced in the literature. PT method is simple, physically intuitive, backward-in-time and mass-conserving algorithm, whose formulation is based on the fundaments of the mechanics of continuous media, in the light of Lagrangian methodology. In present work, we describe new approaches for PT methodology, including an absolutely accurate semi-Lagrangian scheme for 1D problems and a new and attractive discretization for multidimensional problems. In addition, we show that PT method is unconditionally stable.     

A spectral element semi-Lagrangian (SESL) method for the spherical shallow water equations

A spectral element semi-Lagrangian (SESL) method for the shallow water equations on the sphere is presented. The sphere is discretized using a hexahedral grid although any grid imaginable can be used as long as it is comprised of quadrilaterals. The equations are written in Cartesian coordinates to eliminate the pole singularity which plagues the equations in spherical coordinates. In a previous paper [Int. J. Numer. Methods Fluids 35 (2001) 869] we showed how to construct an explicit Eulerian spectral element (SE) model on the sphere; we now extend this work to a semi-Lagrangian formulation. The novelty of the Lagrangian formulation presented is that the high order SE basis functions are used as the interpolation functions for evaluating the values at the Lagrangian departure points. This makes the method not only high order accurate but quite general and thus applicable to unstructured grids and portable to distributed memory computers. The equations are discretized fully implicitly in time in order to avoid having to interpolate derivatives at departure points. By incorporating the Coriolis terms into the Lagrangian derivative, the block LU decomposition of the equations results in a symmetric positive-definite pseudo-Helmholtz operator which we solve using the generalized minimum residual method (GMRES) with a fast projection method [Comput. Methods Appl. Mech. Eng. 163 (1998) 193]. Results for eight test cases are presented to confirm the accuracy and stability of the method. These results show that SESL yields the same accuracy as an Eulerian spectral element semi-implicit (SESI) while allowing for time-steps 10 times as large and being up to 70% more efficient.