Alternating Group Explicit Method For The Numerical Solution Of Non-Linear Singular Two-Point Boundary Value Problems Using A Fourth Order Finite Difference Method

In this paper, we report on the AGE and Newton-AGE iteration methods for the fourth order numerical solution of two point non-linear boundary value problems. Both methods are applicable to problems both in cartesian and polar coordinates and are suitable for use on parallel computers. The convergence analysis of the new method is briefly discussed and the results of numerical experiments presented.     

A Fourth Order Accurate Finite Difference Scheme for the Elastic Wave Equation in Second Order Formulation

We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, (??(x)u _x ) _x . This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of (??(x)u _x ) _x has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.     

Boundary Value Technique for Finding Numerical Solution to Boundary Value Problems for Third Order Singularly Perturbed Ordinary Differential Equations

A class of singularly perturbed two point boundary value problems (BVPs) for third order ordinary differential equations is considered. The BVP is reduced to a weakly coupled system of one first order Ordinary Differential Equation (ODE) with a suitable initial condition and one second order singularly perturbed ODE subject to boundary conditions. In order to solve this system, a computational method is suggested in this paper. This method combines an exponentially fitted finite difference scheme and a classical finite difference scheme. The proposed method is distinguished by the fact that, first we divide the domain of definition of the differential equation into three subintervals called inner and outer regions. Then we solve the boundary value problem over these regions as two point boundary value problems. The terminal boundary conditions of the inner regions are obtained using zero order asymptotic expansion approximation of the solution of the problem. The present method can be extended to system of two equations, of which, one is a first order ODE and the other is a singularly perturbed second order ODE. Examples are presented to illustrate the method.     

一种用于分布参数系统移动边界的自适应剖分数值方法

本文提出通过对具有移动边界分布参数系统中的移动边界的一步预报,自适应生成剖分网格,然后通过系统的焓方程应用有限元方法求解,得到具有移动边界的分布参数系统的数值解.结果表明,这种方法较好地解决了用有限元方法求解该类系统的数值解时遇到的移动边界附近数值解精度与网格剖分过细所导致的计算量过大的矛盾.为具有移动边界的分布参数系统的建模和仿真提供了一种有效的数值计算方法,同时也为研究系统的控制、估计、辨识等问题的数值方法打下了基础.     

A High Order Compact Time/Space Finite Difference Scheme for the Wave Equation with Variable Speed of Sound

We consider fourth order accurate compact schemes, in both space and time, for the second order wave equation with a variable speed of sound. We demonstrate that usually this is much more efficient than lower order schemes despite being implicit and only conditionally stable. Fast time marching of the implicit scheme is accomplished by iterative methods such as conjugate gradient and multigrid. For conjugate gradient, an upper bound on the convergence rate of the iterations is obtained by eigenvalue analysis of the scheme. The implicit discretization technique is such that the spatial and temporal convergence orders can be adjusted independently of each other. In special cases, the spatial error dominates the problem, and then an unconditionally stable second order accurate scheme in time with fourth order accuracy in space is more efficient. Computations confirm the design convergence rate for the inhomogeneous, variable wave speed equation and also confirm the pollution effect for these time dependent problems.     

A Compact Higher Order Finite Difference Method for the Incompressible Navier–Stokes Equations

A numerical method based on compact fourth order finite difference approximations is used for the solution of the incompressible Navier–Stokes equations. Our method is implemented for two dimensional, curvilinear coordinates on orthogonal, staggered grids. Two numerical experiments confirm the theoretically expected order of accuracy.     

Note on a Fourth Order Finite Difference Scheme for the Wave Equation

The linear wave equation is one of the simplest partial differential equations. It has been used as a test equation of hyperbolic systems for different numerical schemes [Richtmyer and Morton (1967); Euvrard (1994); and Lax (1990]. In this short note, a Fourth order finite difference scheme for this equation is proposed and studied. Numerical simulations confirm our theoretical analyses of accuracy and stability condition. It will be interesting to extend the scheme to nonlinear hyperbolic systems.     

High-Order Embedded Finite Difference Schemes for Initial Boundary Value Problems on Time Dependent Irregular Domains

This paper considers a family of spatially discrete approximations, including boundary treatment, to initial boundary value problems in evolving bounded domains. The presented method is based on the Cartesian grid embedded Finite-Difference method, which was initially introduced by Abarbanel and Ditkowski (ICASE Report No. 96-8, 1996; and J. Comput. Phys. 133(2), 1997) and Ditkowski (Ph.D. thesis, Tel Aviv University, 1997), for initial boundary value problems on constant irregular domains. We perform a comprehensive theoretical analysis of the numerical issues, which arise when dealing with domains, whose boundaries evolve smoothly in the spatial domain as a function of time. In this class of problems the moving boundaries are impenetrable with either Dirichlet or Neumann boundary conditions, and should not be confused with the class of moving interface problems such as multiple phase flow, solidification, and the Stefan problem. Unlike other similar works on this class of problems, the resulting method is not restricted to domains of up to 3-D, can achieve higher than 2nd-order accuracy both in time and space, and is strictly stable in semi-discrete settings. The strict stability property of the method also implies, that the numerical solution remains consistent and valid for a long integration time. A complete convergence analysis is carried in semi-discrete settings, including a detailed analysis for the implementation of the diffusion equation. Numerical solutions of the diffusion equation, using the method for a 2nd and a 4th-order of accuracy are carried out in one dimension and two dimensions respectively, which demonstrates the efficacy of the method. This research was supported by the Israel Science Foundation (grant No. 1362/04).     

Methods to Construct the Accurate Difference Scheme for a Differential Equation of Order 4

We construct accurate difference schemes for fourth-order equations with variable coefficients. To this end, we use the explicit solutions of the Cauchy problem.     

A Lagrangian Scheme for the Solution of Nonlinear Diffusion Equations Using Moving Simplex Meshes

A Lagrangian numerical scheme for solving nonlinear degenerate Fokker–Planck equations in space dimensions \(d\ge 2\) is presented. It applies to a large class of nonlinear diffusion equations, whose dynamics are driven by internal energies and given external potentials, e.g. the porous medium equation and the fast diffusion equation. The key ingredient in our approach is the gradient flow structure of the dynamics. For discretization of the Lagrangian map, we use a finite subspace of linear maps in space and a variational form of the implicit Euler method in time. Thanks to that time discretisation, the fully discrete solution inherits energy estimates from the original gradient flow, and these lead to weak compactness of the trajectories in the continuous limit. Consistency is analyzed in the planar situation, \(d=2\). A variety of numerical experiments for the porous medium equation indicates that the scheme is well-adapted to track the growth of the solution’s support.     

Domain Decomposition Spectral Method for Mixed Inhomogeneous Boundary Value Problems of High Order Differential Equations on Unbounded Domains

In this paper, we develop domain decomposition spectral method for mixed inhomogeneous boundary value problems of high order differential equations defined on unbounded domains. We introduce an orthogonal family of new generalized Laguerre functions, with the weight function x ^?? , ?? being any real number. The corresponding quasi-orthogonal approximation and Gauss-Radau type interpolation are investigated, which play important roles in the related spectral and collocation methods. As examples of applications, we propose the domain decomposition spectral methods for two fourth order problems, and the spectral method with essential imposition of boundary conditions. The spectral accuracy is proved. Numerical results demonstrate the effectiveness of suggested algorithms.     

A Single-Step Characteristic-Curve Finite Element Scheme of Second Order in Time for the Incompressible Navier-Stokes Equations

In this paper we present a new single-step characteristic-curve finite element scheme of second order in time for the nonstationary incompressible Navier-Stokes equations. After supplying correction terms in the variational formulation, we prove that the scheme is of second order in time. The convergence rate of the scheme is numerically recognized by computational results.     

A Supra-Convergent Finite Difference Scheme for the Poisson and Heat Equations on Irregular Domains and Non-Graded Adaptive Cartesian Grids

We present finite difference schemes for solving the variable coefficient Poisson and heat equations on irregular domains with Dirichlet boundary conditions. The computational domain is discretized with non-graded Cartesian grids, i.e., grids for which the difference in size between two adjacent cells is not constrained. Refinement criteria is based on proximity to the irregular interface such that cells with the finest resolution is placed on the interface. We sample the solution at the cell vertices (nodes) and use quadtree (in 2D) or octree (in 3D) data structures as efficient means to represent the grids. The boundary of the irregular domain is represented by the zero level set of a signed distance function. For cells cut by the interface, the location of the intersection point is found by a quadratic fitting of the signed distance function, and the Dirichlet boundary value is obtained by quadratic interpolation. Instead of using ghost nodes outside the interface, we use directly this intersection point in the discretization of the variable coefficient Laplacian. These methods can be applied in a dimension-by-dimension fashion, producing schemes that are straightforward to implement. Our method combines the ability of adaptivity on quadtrees/octrees with a quadratic treatment of the Dirichlet boundary condition on the interface. Numerical results in two and three spatial dimensions demonstrate second-order accuracy for both the solution and its gradients in the L ¹ and L ^∞ norms.