Comparison of two Eulerian solvers for the four-dimensional Vlasov equation: Part I

This paper presents two methods for solving the four-dimensional Vlasov equation on a grid of the phase space. The two methods are based on the semi-Lagrangian method which consists in computing the distribution function at each grid point by following the characteristic curves ending there. The first method reconstructs the distribution function using local splines which are well suited for a parallel implementation. The second method is adaptive using wavelets interpolation: only a subset of the grid points are conserved to manage data locality. Numerical results are presented in the second part.     

Low Dimensional Born-Infeld Equations Coupled with a Collisionless Matter Model

We consider the Born-Infeld nonlinear electromagnetic field equations and study its Cauchy problem in the case that the Vlasov equation is considered as a matter model. In the present paper, the Vlasov equation is considered on the so-called one and one-half dimensional phase space, and in consequence the Born-Infeld equations are reduced to a quasilinear hyperbolic system with two unknowns. A transformation is introduced in order to make the field equations easy to handle, and suitable assumptions are made on initial data so that the nonlinearity of the field is controlled.     

Compactly supported wavelet and the numerical solution of the vlasov equation

A new scheme for solving the Vlasov equation using a compactly supported wavelets basis is proposed. We use a numerical method which minimizes the numerical diffusion and conserves a reasonable time computing cost. So we introduce a representation in a compactly supported wavelet of the derivative operator. This method makes easy and simple the computation of the coefficients of the matrix representing the operator. This allows us to solve the two equations which result from the splitting technique of the main Vlasov equation. Some numerical results are exposed using different numbers of wavelets.     

Wavelet based schemes for linear advection-dispersion equation

In this paper, two wavelet based adaptive solvers are developed for linear advection-dispersion equation. The localization properties and multilevel structure of the wavelets in the physical space are used for adaptive computational methods for solution of equation which exhibit both smooth and shock-like behaviour. The first framework is based on wavelet-Galerkin and the second is based on multiscale decomposition of finite element method. Coiflet wavelet filter is incorporated in both the methods. The main advantage of both the adaptive methods is the elimination of spurious oscillations at very high Peclet number.     

一个求解约束非线性优化问题的微分方程方法

本文构造的求解非线性优化问题的微分方程方法包括两个微分方程系统,第一个系统基于问题函数的一阶信息,第二个系统基于二阶信息．这两个系统具有性质:非线性优化问题的局部最优解是它们的渐近稳定的平衡点,并且初始点是可行点时,解轨迹都落于可行域中．我们证明了两个微分方程系统的离散迭代格式的收敛性定理和基于第二个系统的离散迭代格式的局部二次收敛性质．还给出了基于两个系统的离散迭代方法的数值算例,数值结果表明基于二阶信息的微分方程方法速度更快．     

An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented.     

Two finite difference schemes for the phase field crystal equation

The phase field crystal(PFC) model is a nonlinear evolutionary equation that is of sixth order in space.In the first part of this work,we derive a three level linearized difference scheme,which is then proved to be energy stable,uniquely solvable and second order convergent in L_2 norm by the energy method combining with the inductive method.In the second part of the work,we analyze the unique solvability and convergence of a two level nonlinear difference scheme,which was developed by Zhang et al.in 2013.Some numerical results with comparisons are provided.     

Relaxation approximation to bed-load sediment transport

In this work we propose and apply a numerical method based on finite volume relaxation approximation for computing the bed-load sediment transport in shallow water flows, in one and two space dimensions. The water flow is modeled by the well-known nonlinear shallow water equations which are coupled with a bed updating equation. Using a relaxation approximation, the nonlinear set of equations (and for two different formulations) is transformed to a semilinear diagonalizable problem with linear characteristic variables. A second order MUSCL-TVD method is used for the advection stage while an implicit–explicit Runge–Kutta scheme solves the relaxation stage. The main advantages of this approach are that neither Riemann problem solvers nor nonlinear iterations are required during the solution process. For the two different formulations, the applicability and effectiveness of the presented scheme is verified by comparing numerical results obtained for several benchmark test problems.     

Simulation of Multiphase Flows with Strong Shocks and Density Variations

The system of extended Euler type hyperbolic equations is considered to describe a two-phase compressible flow. A numerical scheme for computing multi-component flows is then examined. The numerical approach is based on the mathematical model that considers interfaces between fluids as numerically diffused zones. The hyperbolic problem is tackled using a high resolution HLLC scheme on a fixed Eulerian mesh. The global set of conservative equations (mass, momentum and energy) for each phase is closed with a general two parameters equation of state for each constituent. The performance of various variants of a diffuse interface method is carefully verified against a comprehensive suite of numerical benchmark test cases in one and two space dimensions. The studied benchmark cases are divided into two categories: idealized tests for which exact solutions can be generated and tests for which the equivalent numerical results could be obtained using different approaches. The ability to simulate the Richtmyer-Meshkov instabilities, which are generated when a shock wave impacts an interface between two different fluids, is considered as a major challenge for the present numerical techniques. The study presents the effect of density ratio of constituent fluids on the resolution of an interface and the ability to simulate Richtmyer-Meshkov instabilities by various variants of diffuse interface methods. (© 2011 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A commuting-vector-field approach to some dispersive estimates

We prove the pointwise decay of solutions to three linear equations: (1) the transport equation in phase space generalizing the classical Vlasov equation, (2) the linear Schrödinger equation, (3) the Airy (linear KdV) equation. The usual proofs use explicit representation formulae, and either obtain \(L^1\)—\(L^\infty \) decay through directly estimating the fundamental solution in physical space or by studying oscillatory integrals coming from the representation in Fourier space. Our proof instead combines “vector field” commutators that capture the inherent symmetries of the relevant equations with conservation laws for mass and energy to get space–time weighted energy estimates. Combined with a simple version of Sobolev’s inequality this gives pointwise decay as desired. In the case of the Vlasov and Schrödinger equations, we can recover sharp pointwise decay; in the Schrödinger case we also show how to obtain local energy decay as well as Strichartz-type estimates. For the Airy equation we obtain a local energy decay that is almost sharp from the scaling point of view, but nonetheless misses the classical estimates by a gap. This work is inspired by the work of Klainerman on \(L^2\)—\(L^\infty \) decay of wave equations, as well as the recent work of Fajman, Joudioux, and Smulevici on decay of mass distributions for the relativistic Vlasov equation.     

偏微分方程的区间小波自适应精细积分法

利用插值小波理论构造了拟Shannon区间小波,并结合外推法给出了一种求解非线性常微分方程组的时间步长自适应精细积分法,在此基础上构造了求解非线性偏微分方程的区间小波自适应精细积分法（AIWPIM）．数值结果表明,该方法在计算精度上优于将小波和四阶Runge-Kutta法组合得到的偏微分方程的数值求解方法,而计算量则相差不大．该文方法通过Burgers方程给出,但适用于一般情形．     

Numerical resolution of the Vlasov equation for the Hamiltonian Mean-Field model

We present in this paper detailed numerical Vlasov simulations of the Hamiltonian Mean-Field model. This model is used as a representative of the class of systems under long-range interactions. We check existing results on the stability of the homogeneous situation and analyze numerical properties of the semi-Lagrangian time-split algorithm for solving the Vlasov equation. We also detail limitations due to finite resolution of the method.     

A Sequential Least Squares Method for Elliptic Equations in Non-Divergence Form

We develop a new least squares method for solving the second-order elliptic equations in non-divergence form. Two least-squares-type functionals are proposed for solving the equation in two sequential steps. We first obtain a numerical approximation to the gradient in a piecewise irrotational polynomial space. Then together with the numerical gradient, we seek a numerical solution of the primitive variable in the continuous Lagrange finite element space. The variational setting naturally provides an a posteriori error which can be used in an adaptive refinement algorithm. The error estimates under the $L^2$ norm and the energy norm for both two unknowns are derived. By a series of numerical experiments, we verify the convergence rates and show the efficiency of the adaptive algorithm.     

On global symmetric solutions to the relativistic Vlasov–Poisson equation in three space dimensions

A collisionless plasma is modelled by the Vlasov–Maxwell system. In the presence of very large velocities, relativistic corrections are meaningful. When magnetic effects are ignored this formally becomes the relativistic Vlasov–Poisson equation. The initial datum for the phase space density ƒ₀(x, v) is assumed to be sufficiently smooth, non‐negative and cylindrically symmetric. If the (two‐dimensional) angular momentum is bounded away from zero on the support of ƒ₀(x, v), it is shown that a smooth solution to the Cauchy problem exists for all times. Copyright © 2001 John Wiley & Sons, Ltd.     

Numerical treatment of two‐parameter singularly perturbed parabolic convection diffusion problems with non‐smooth data

In the present work, we consider a parabolic convection‐diffusion‐reaction problem where the diffusion and convection terms are multiplied by two small parameters, respectively. In addition, we assume that the convection coefficient and the source term of the partial differential equation have a jump discontinuity. The presence of perturbation parameters leads to the boundary and interior layers phenomena whose appropriate numerical approximation is the main goal of this paper. We have developed a uniform numerical method, which converges almost linearly in space and time on a piecewise uniform space adaptive Shishkin‐type mesh and uniform mesh in time. Error tables based on several examples show the convergence of the numerical solutions. In addition, several numerical simulations are presented to show the effectiveness of resolving layer behavior and their locations.     

On the Numerical Solution to a Nonlinear Wave Equation Associated with the First Painlev′e Equation: an Operator-Splitting Approach

The main goal of this article is to discuss the numerical solution to a nonlinear wave equation associated with the first of the celebrated Painlevé transcendent ordinary differential equations. In order to solve numerically the above equation, whose solutions blow up in finite time, the authors advocate a numerical methodology based on the Strang’s symmetrized operator-splitting scheme. With this approach, one can decouple nonlinearity and differential operators, leading to the alternate solution at every time step of the equation as follows: (i) The first Painlevé ordinary differential equation, (ii) a linear wave equation with a constant coefficient. Assuming that the space dimension is two, the authors consider a fully discrete variant of the above scheme, where the space-time discretization of the linear wave equation sub-steps is achieved via a Galerkin/finite element space approximation combined with a second order accurate centered time discretization scheme. To handle the nonlinear sub-steps, a second order accurate centered explicit time discretization scheme with adaptively variable time step is used, in order to follow accurately the fast dynamic of the solution before it blows up. The results of numerical experiments are presented for different coefficients and boundary conditions. They show that the above methodology is robust and describes fairly accurately the evolution of a rather “violent” phenomenon.     

On Numerical Landau Damping for Splitting Methods Applied to the Vlasov–HMF Model

We consider time discretizations of the Vlasov–HMF (Hamiltonian mean-field) equation based on splitting methods between the linear and nonlinear parts. We consider solutions starting in a small Sobolev neighborhood of a spatially homogeneous state satisfying a linearized stability criterion (Penrose criterion). We prove that the numerical solutions exhibit a scattering behavior to a modified state, which implies a nonlinear Landau damping effect with polynomial rate of damping. Moreover, we prove that the modified state is close to the continuous one and provide error estimates with respect to the time step size.     

Analytical approximate solutions of Riesz fractional diffusion equation and Riesz fractional advection–dispersion equation involving nonlocal space fractional derivatives

In this paper, we consider the analytical solutions of fractional partial differential equations (PDEs) with Riesz space fractional derivatives on a finite domain. Here we considered two types of fractional PDEs with Riesz space fractional derivatives such as Riesz fractional diffusion equation (RFDE) and Riesz fractional advection–dispersion equation (RFADE). The RFDE is obtained from the standard diffusion equation by replacing the second‐order space derivative with the Riesz fractional derivative of order α∈(1,2]. The RFADE is obtained from the standard advection–dispersion equation by replacing the first‐order and second‐order space derivatives with the Riesz fractional derivatives of order β∈(0,1] and of order α∈(1,2] respectively. Here the analytic solutions of both the RFDE and RFADE are derived by using modified homotopy analysis method with Fourier transform. Then, we analyze the results by numerical simulations, which demonstrate the simplicity and effectiveness of the present method. Here the space fractional derivatives are defined as Riesz fractional derivatives. Copyright © 2015 John Wiley & Sons, Ltd.     

Vlasov–Poisson: The waterbag method revisited

We revisit, with a view to refinement and generalization, the elegant waterbag method for the numerical treatment of Vlasov–Poisson equations. In this method, the phase space is decomposed into patches of constant density, and by exploiting Liouville’s theorem, the dynamics is reduced to the evolution of the boundary of these patches (waterbags). We follow the boundary using an adaptive, oriented polygon, and recover the force by circulating along this polygon. We discuss sampling of initial conditions with a set of oriented isocontours, and propose a new refinement procedure for accurate rendering of the stretching and folding polygon. Time evolution is naturally undertaken with symplectic algorithms. Tools, initially developed for systems of self-gravitating sheets, generalize naturally to spherically symmetric systems. We conclude with examples of both cases.     

Hamiltonian restriction of Vlasov equations to rotating isopycnic and isentropic two-layer equations

A direct link between a Vlasov equation and the equations of motion of a rotating fluid with an effective pressure depending only on a pseudo-density is illustrated. In this direct link, the resulting fluid equations necessarily appear in flux conservative form when there are no topographical and rotational terms. In contrast, multilayer isopycnic and isentropic equations used in atmosphere and ocean dynamics, in the absence of topographical and rotational terms, cannot be brought into a conservative flux form, and, hence, cannot be derived directly from the Vlasov equations. Another route is explored, therefore: deriving the Hamiltonian formulation of the two-layer isopycnic and isentropic equations as a restriction from a Hamiltonian formulation of two decoupled Vlasov equations. The work is motivated by our search for energy-preserving or even Hamiltonian (kinetic) numerical schemes.