High-order maximum-principle-preserving and positivity-preserving weighted compact nonlinear schemes for hyperbolic conservation laws

In this paper,the maximum-principle-preserving(MPP)and positivitypreserving(PP)flux limiting technique will be generalized to a class of high-order weighted compact nonlinear schemes(WCNSs)for scalar conservation laws and the compressible Euler systems in both one and two dimensions.The main idea of the present method is to rewrite the scheme in a conservative form,and then define the local limiting parameters via case-by-case discussion.Smooth test problems are presented to demonstrate that the proposed MPP/PP WCNSs incorporating a third-order Runge-Kutta method can attain the desired order of accuracy.Other test problems with strong shocks and high pressure and density ratios are also conducted to testify the performance of the schemes.

Immersed Interface Method for Fokker-Planck Equation with Discontinuous Drift

The Immersed Interface Method (IIM) is derived to solve the corresponding Fokker-Planck equation of Brownian motion with pure dry friction, which is one of the simplest models of piecewise-smooth stochastic systems. The IIM is capable of treating a discontinuity in the drift of Fokker-Planck equation and it is readily extended to the dry and viscous friction model. Analytic results of the considered model are used to confirm the effectiveness and design accuracy of the scheme.

SOLVING NONLINEAR DELAY-DIFFERENTIAL-ALGEBRAIC EQUATIONS WITH SINGULAR PERTURBATION VIA BLOCK BOUNDARY VALUE METHODS

Block boundary value methods(BBVMs)are extended in this paper to obtain the numerical solutions of nonlinear delay-differential-algebraic equations with singular perturbation(DDAESP).It is proved that the extended BBVMs in some suitable conditions are globally stable and can obtain a unique exact solution of the DDAESP.Besides,whenever the classic Lipschitz conditions are satisfied,the extended BBVMs are preconsistent and pth order consistent.Moreover,through some numerical examples,the correctness of the theoretical results and computational validity of the extended BBVMs is further confirmed.

Fourth-Order Structure-Preserving Method for the Conservative Allen-Cahn Equation

We propose a class of up to fourth-order maximum-principle-preserving and mass-conserving schemes for the conservative Allen-Cahn equation equipped with a non-local Lagrange multiplier.Based on the second-order finite-difference semidiscretization in the spatial direction,the integrating factor Runge-Kutta schemes are applied in the temporal direction.Theoretical analysis indicates that the proposed schemes conserve mass and preserve the maximum principle under reasonable time step-size restriction,which is independent of the space step size.Finally,the theoretical analysis is verified by several numerical examples.

二维非线性Schr?dinger方程的两类局部守恒算法

本文针对二维非线性Schr?dinger方程,提出两类局部守恒算法.不需要考虑边界条件,即可保持任意时空区域上相应的局部能量守恒律和局部动量守恒律.在合适的边界条件下,它们能自然地保持电荷、全局能量或全局动量守恒律.本文同时对算法进行了守恒分析和误差分析.在数值实验部分,本文构造了类似的多辛Preissman算法进行比较,数值结果验证了其长时间计算的优势.

A Maximum-Principle-Preserving Third Order Finite Volume SWENO Scheme on Unstructured Triangular Meshes

We modify the construction of the third order finite volume WENO scheme on triangular meshes and present a simplified WENO(SWENO)scheme.The novelty of the SWENO scheme is the less complexity and lower computational cost when deciding the smoothest stencil through a simple mechanism.The LU decomposition with iterative refinement is adopted to implement ill-conditioned interpolation matrices and improves the stability of the SWENOscheme efficiently.Besides,a scaling technique is used to circument the growth of condition numbers as mesh refined.However,weak oscillations still appear when the SWENO scheme deals with complex low density equations.In order to guarantee the maximum-principle-preserving(MPP)property,we apply a scaling limiter to the reconstruction polynomial without the loss of accuracy.A novel procedure is designed to prove this property theoretically.Finally,numerical examples for one-and two-dimensional problems are presented to verify the good performance,maximum principle preserving,essentially non-oscillation and high resolution of the proposed scheme.

A Well-Balanced Weighted Compact Nonlinear Scheme for Pre-Balanced Shallow Water Equations

It is well known that developing well-balanced schemes for the balance laws is useful for reducing numerical errors.In this paper,a well-balanced weighted compact nonlinear scheme(WCNS)is proposed for shallow water equations in prebalanced forms.The scheme is proved to be well-balanced provided that the source term is treated appropriately as the advection term.Some numerical examples in oneand two-dimensions are also presented to demonstrate the well-balanced property,high order accuracy and good shock capturing capability of the proposed scheme.

Multi-SymplecticWavelet Collocation Method for Maxwell’s Equations

In this paper,we develop a multi-symplectic wavelet collocation method for three-dimensional(3-D)Maxwell’s equations.For the multi-symplectic formulation of the equations,wavelet collocation method based on autocorrelation functions is applied for spatial discretization and appropriate symplectic scheme is employed for time integration.Theoretical analysis shows that the proposed method is multi-symplectic,unconditionally stable and energy-preserving under periodic boundary conditions.The numerical dispersion relation is investigated.Combined with splitting scheme,an explicit splitting symplectic wavelet collocation method is also constructed.Numerical experiments illustrate that the proposed methods are efficient,have high spatial accuracy and can preserve energy conservation laws exactly.

Robust and Quality Boundary Constrained Tetrahedral Mesh Generation

A novel method for boundary constrained tetrahedral mesh generation is proposed based on Advancing Front Technique(AFT)and conforming Delaunay triangulation.Given a triangulated surface mesh,AFT is firstly applied to mesh several layers of elements adjacent to the boundary.The rest of the domain is then meshed by the conforming Delaunay triangulation.The non-conformal interface between two parts of meshes are adjusted.Mesh refinement and mesh optimization are then preformed to obtain a more reasonable-sized mesh with better quality.Robustness and quality of the proposed method is shown.Convergence proof of each stage as well as the whole algorithm is provided.Various numerical examples are included as well as the quality of the meshes.

A Splitting Method for the Degasperis-Procesi Equation Using an Optimized WENO Scheme and the Fourier Pseudospectral Method

The Degasperis-Procesi(DP)equation is split into a system of a hyperbolic equation and an elliptic equation.For the hyperbolic equation,we use an optimized finite difference weighted essentially non-oscillatory(OWENO)scheme.New smoothness measurement is presented to approximate the typical shockpeakon structure in the solution to the DP equation,which evidently reduces the dissipation arising from discontinuities simultaneously removing nonphysical oscillations.For the elliptic equation,the Fourier pseudospectral method(FPM)is employed to discretize the high order derivative.Due to the combination of the WENO reconstruction and FPM,the splitting method shows an excellent performance in capturing the formation and propagation of shockpeakon solutions.The numerical simulations for different solutions of the DP equation are conducted to illustrate the high accuracy and capability of the method.

Novel Conformal Structure-Preserving Algorithms for Coupled Damped Nonlinear Schr odinger System

This paper introduces two novel conformal structure-preserving algorithms for solving the coupled damped nonlinear Schr¨odinger(CDNLS)system,which are based on the conformal multi-symplectic Hamiltonian formulation and its conformal conservation laws.The proposed algorithms can preserve corresponding conformal multi-symplectic conservation lawand conformalmomentum conservation lawin any local time-space region,respectively.Moreover,it is further shown that the algorithms admit the conformal charge conservation law,and exactly preserve the dissipation rate of charge under appropriate boundary conditions.Numerical experiments are presented to demonstrate the conformal properties and effectiveness of the proposed algorithms during long-time numerical simulations and validate the analysis.

Explicit Multi-Symplectic Splitting Methods for the Nonlinear Dirac Equation

In this paper,we propose two new explicit multi-symplectic splitting methods for the nonlinear Dirac(NLD)equation.Based on its multi-symplectic formulation,the NLD equation is split into one linear multi-symplectic system and one nonlinear infinite Hamiltonian system.Then multi-symplectic Fourier pseudospectral method and multi-symplectic Preissmann scheme are employed to discretize the linear subproblem,respectively.And the nonlinear subsystem is solved by a symplectic scheme.Finally,a composition method is applied to obtain the final schemes for the NLD equation.We find that the two proposed schemes preserve the total symplecticity and can be solved explicitly.Numerical experiments are presented to show the effectiveness of the proposed methods.

Novel Conservative Methods for Schrödinger Equations with Variable Coefficients over Long Time

In this paper,we propose a wavelet collocation splitting(WCS)method,and a Fourier pseudospectral splitting(FPSS)method as comparison,for solving onedimensional and two-dimensional Schrödinger equations with variable coefficients in quantum mechanics.The two methods can preserve the intrinsic properties of original problems as much as possible.The splitting technique increases the computational efficiency.Meanwhile,the error estimation and some conservative properties are investigated.It is proved to preserve the charge conservation exactly.The global energy and momentum conservation laws can be preserved under several conditions.Numerical experiments are conducted during long time computations to show the performances of the proposed methods and verify the theoretical analysis.