Numerical methods for fourth‐order nonlinear elliptic boundary value problems

The aim of this article is to present several computational algorithms for numerical solutions of a nonlinear finite difference system that represents a finite difference approximation of a class of fourth‐order elliptic boundary value problems. The numerical algorithms are based on the method of upper and lower solutions and its associated monotone iterations. Three linear monotone iterative schemes are given, and each iterative scheme yields two sequences, which converge monotonically from above and below, respectively, to a maximal solution and a minimal solution of the finite difference system. This monotone convergence property leads to upper and lower bounds of the solution in each iteration as well as an existence‐comparison theorem for the finite difference system. Sufficient conditions for the uniqueness of the solution and some techniques for the construction of upper and lower solutions are obtained, and numerical results for a two‐point boundary‐value problem with known analytical solution are given. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17:347–368, 2001     

A spline collocation approach for the numerical solution of a generalized nonlinear Klein-Gordon equation

In this paper, a finite element collocation approach using cubic B-splines is employed for the numerical solution of a generalized form of the nonlinear Klein-Gordon equation. The efficiency of the method is tested on a number of examples that represent special cases of the extended equation including the sine-Gordon equation. The numerical results are compared with existing numerical and analytic solutions and the outcomes confirm that the scheme yields accurate and reliable results even when few nodes are used at the time levels.     

A parallel well-balanced finite volume method for shallow water equations with topography on the cubed-sphere

A finite volume scheme for the global shallow water model on the cubed-sphere mesh is proposed and studied in this paper. The new cell-centered scheme is based on Osher’s Riemann solver together with a high-order spatial reconstruction. On each patch interface of the cubed-sphere only one layer of ghost cells is needed in the scheme and the numerical flux is calculated symmetrically across the interface to ensure the numerical conservation of total mass. The discretization of the topographic term in the equation is properly modified in a well-balanced manner to suppress spurious oscillations when the bottom topography is non-smooth. Numerical results for several test cases including a steady-state nonlinear geostrophic flow and a zonal flow over an isolated mountain are provided to show the flexibility of the scheme. Some parallel implementation details as well as some performance results on a parallel supercomputer with more than one thousand processor cores are also provided.     

Numerical study of Sivashinsky equation using a splitting scheme based on Crank‐Nicolson method

The mathematical modeling of a planar solid‐liquid interface in the solidification of a dilute binary alloy is formulating by one of nonintegrable, nonlinear evolution equation known as Sivashinsky equation. In the first part of this paper, the mathematical modeling of Sivashinsky equation is briefly discussed. Since, the exact solutions of this equation is yet unknown, obtaining its numerical solution plays an important role to simulate its behavior. Therefore, in the second part, a second‐order splitting finite difference scheme, based on Crank‐Nicolson method, is investigated to approximate the solution of the Sivashinsky equation with homogeneous boundary conditions. We prove the solvability of the present scheme and establish the error estimate of the numerical scheme.     

Finite difference reaction diffusion equations with nonlinear boundary conditions

This article is concerned with numerical solutions of finite difference systems of reaction diffusion equations with nonlinear internal and boundary reaction functions. The nonlinear reaction functions are of general form and the finite difference systems are for both time-dependent and steady-state problems. For each problem a unified system of nonlinear equations is treated by the method of upper and lower solutions and its associated monotone iterations. This method leads to a monotone iterative scheme for the computation of numerical solutions as well as an existence-comparison theorem for the corresponding finite difference system. Special attention is given to the dynamical property of the time-dependent solution in relation to the steady-state solutions. Application is given to a heat-conduction problem where a nonlinear radiation boundary condition obeying the Boltzmann law of cooling is considered. This application demonstrates a bifurcation property of two steady-state solutions, and determines the dynamic behavior of the time-dependent solution. Numerical results for the heat-conduction problem, including a test problem with known analytical solution, are presented to illustrate the various theoretical conclusions. © 1995 John Wiley & Sons, Inc.     

Two-Grid Algorithm of $H^{1}$-Galerkin Mixed Finite Element Methods for Semilinear Parabolic Integro-Differential Equations

In this paper, we present a two-grid discretization scheme for semilinear parabolic integro-differential equations by $H^{1}$-Galerkin mixed finite element methods. We use the lowest order Raviart-Thomas mixed finite elements and continuous linear finite element for spatial discretization, and backward Euler scheme for temporal discretization. Firstly, a priori error estimates and some superclose properties are derived. Secondly, a two-grid scheme is presented and its convergence is discussed. In the proposed two-grid scheme, the solution of the nonlinear system on a fine grid is reduced to the solution of the nonlinear system on a much coarser grid and the solution of two symmetric and positive definite linear algebraic equations on the fine grid and the resulting solution still maintains optimal accuracy. Finally, a numerical experiment is implemented to verify theoretical results of the proposed scheme. The theoretical and numerical results show that the two-grid method achieves the same convergence property as the one-grid method with the choice $h=H^2$.     

IIM-based ADI finite difference scheme for nonlinear convection–diffusion equations with interfaces

Based on Li’s immersed interface method (IIM), an ADI-type finite difference scheme is proposed for solving two-dimensional nonlinear convection–diffusion interface problems on a fixed cartesian grid, which is unconditionally stable and converges with two-order accuracy in both time and space in maximum norm. Correction terms are added to the right-hand side of standard ADI scheme at irregular points. The nonlinear convection terms are treated by Adams–Bashforth method, without affecting the stability of difference schemes. A new method for computing the correction terms is developed, in which the Adams–Bashforth method is employed. Thus we can get an explicit approximation for the computation of corrections, when the jump condition is solution-dependent. Three numerical experiments are displayed and analyzed. The numerical results show good agreement with the exact solutions and confirm the convergence order.     

Finite element simulations of window Josephson junctions

This paper deals with the numerical simulation of the steady state two dimensional window Josephson junctions by finite element method. The model is represented by a sine-Gordon type composite PDE problem. Convergence and error analysis of the finite element approximation for this semilinear problem are presented. An efficient and reliable Newton-preconditioned conjugate gradient algorithm is proposed to solve the resulting nonlinear discrete system. Regular solution branches are computed using a simple continuation scheme. Numerical results associated with interesting physical phenomena are reported. Interface relaxation methods, which by taking advantage of special properties of the composite PDE, can further reduce the overall computational cost are proposed. The implementation and the associated numerical experiments of a particular interface relaxation scheme are also presented and discussed.     

CONVERGENCE OF THE CRANK-NICOLSON/NEWTON SCHEME FOR NONLINEAR PARABOLIC PROBLEM

In this paper, the Crank-Nicolson/Newton scheme for solving numerically secondorder nonlinear parabolic problem is proposed. The standard Galerkin finite element method based on P2 conforming elements is used to the spatial discretization of the problem and the Crank-Nicolson/Newton scheme is applied to the time discretization of the resulted finite element equations. Moreover, assuming the appropriate regularity of the exact solution and the finite element solution, we obtain optimal error estimates of the fully discrete CrankNicolson/Newton scheme of nonlinear parabolic problem. Finally, numerical experiments are presented to show the efficient performance of the proposed scheme.     

On the numerical solution of nonlinear Burgers’-type equations using meshless method of lines

In this paper, a meshless method of lines (MMOL) is proposed for the numerical solution of nonlinear Burgers’-type equations. This technique does not require a mesh in the problem domain, and only a set of scattered nodes provided by initial data is required for the solution of the problem using some radial basis functions (RBFs). The scheme is tested for various examples. The results obtained by this method are compared with the exact solutions and some earlier work.     

Conservative finite difference schemes for the generalized Zakharov–Kuznetsov equations

This paper is concerned with the construction of conservative finite difference schemes by means of discrete variational method for the generalized Zakharov–Kuznetsov equations and the numerical solvability of the two-dimensional nonlinear wave equations. A finite difference scheme is proposed such that mass and energy conservation laws associated with the generalized Zakharov–Kuznetsov equations hold. Our arguments are based on the procedure that D. Furihata has recently developed for real-valued nonlinear partial differential equations. Numerical results are given to confirm the accuracy as well as validity of the numerical solutions and then exhibit remarkable nonlinear phenomena of the interaction and behavior of pulse wave solutions.     

A new predictor-corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black-Scholes partial differential equation, a predictor-corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.     

Semi-analytical method for solving nonlinear heat diffusion problems in spherical medium

A semi-analytical methodology, based on the finite integral transform technique, is proposed to solve the heat diffusion problem in a spherical medium subject to nonlinear boundary conditions due to radiation exchange at the interface according to the fourth power law. The method proceeds by treating the nonlinearity term in the boundary condition as a source in the differential equation and keeping other conditions unchanged. The results obtained from this semi-analytical solutions are compared with those obtained from a numerical solution developed using an explicit finite difference method, which showed very good agreement.     

A finite difference scheme for the nonlinear time-fractional partial integro-differential equation

In this paper, a finite difference scheme is proposed for solving the nonlinear time-fractional integro-differential equation. This model involves two nonlocal terms in time, ie, a Caputo time-fractional derivative and an integral term with memory. The existence of numerical solutions is shown by the Leray-Schauder theorem. And we obtain the discrete L₂ stability and convergence with second order in time and space by the discrete energy method. Then the uniqueness of numerical solutions is derived. Moreover, an iterative algorithm is designed for solving the derived nonlinear system. Numerical examples are presented to validate the theoretical findings and the efficiency of the proposed algorithm.     

Second Order Unconditionally Stable and Convergent Linearized Scheme for a Fluid-Fluid Interaction Model

In this paper, a fully discrete finite element scheme with second-order temporal accuracy is proposed for a fluid-fluid interaction model, which consists of two Navier-Stokes equations coupled by a linear interface condition. The proposed fully discrete scheme is a combination of a mixed finite element approximation for spatial discretization, the second-order backward differentiation formula for temporal discretization, the second-order Gear's extrapolation approach for the interface terms and extrapolated treatments in linearization for the nonlinear terms. Moreover, the unconditional stability is established by rigorous analysis and error estimate for the fully discrete scheme is also derived. Finally, some numerical experiments are carried out to verify the theoretical results and illustrate the accuracy and efficiency of the proposed scheme.     

The IIM in polar coordinates and its application to electro capacitance tomography problems

In a numerical study of the electric field inside an Electro Capacitance Tomography (ECT) device and other applications, often a Poisson equation with a discontinuous coefficient needs to be solved in polar coordinates. This paper is devoted to the Immersed Interface Method (IIM) in polar coordinates and the application to the solution of the electric potential inside an ECT device. The numerical algorithm is based on a finite difference discretization on a uniform polar coordinates grid. The finite difference scheme is modified at grid points near and on the interface across which the coefficient is discontinuous so that the natural jump conditions are satisfied. The algorithm and analysis here is one step forward in applying the IIM for 3D problems in axisymmetric situations or in the spherical coordinates. Numerical examples against exact solutions and the application to ECT problems are also presented.     

Numerical solution of the nonlinear Klein–Gordon equation using radial basis functions

The nonlinear Klein–Gordon equation is used to model many nonlinear phenomena. In this paper, we propose a numerical scheme to solve the one-dimensional nonlinear Klein–Gordon equation with quadratic and cubic nonlinearity. Our scheme uses the collocation points and approximates the solution using Thin Plate Splines (TPS) radial basis functions (RBF). The implementation of the method is simple as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme.     

ON NUMEROV SCHEME FOR NONLINEAR TWO-POINTS BOUNDARY VALUE PROBLEM

1.IntroductionInstudyingsomeproblemsarisinginelectromagnetism,biology)astronomy,bound-arylayerandothertopics,weoftenmeetnonlineartwo--pointsboundaryproblem,i.e.,findingyECo[0,11nC2(0,1)suchthatwhereor,garecertainconstants,andf(x,z)EC'(0,1)xC'(--co,co).Undersomeconditionsonf(x,z),wecanusetheframeworkof[1]toinvestigatetheexistenceanduniquenessofitssolutions.Alsotherearealotofliteratureconcerningitsnumericalsolutio.s[2--'].Inparticular,N..ero.15]proposedafamousfinitedifferenceschemewiththeaccu…     

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.