Maximum norm error estimates of efficient difference schemes for second-order wave equations

The three-level explicit scheme is efficient for numerical approximation of the second-order wave equations. By employing a fourth-order accurate scheme to approximate the solution at first time level, it is shown that the discrete solution is conditionally convergent in the maximum norm with the convergence order of two. Since the asymptotic expansion of the difference solution consists of odd powers of the mesh parameters (time step and spacings), an unusual Richardson extrapolation formula is needed in promoting the second-order solution to fourth-order accuracy. Extensions of our technique to the classical ADI scheme also yield the maximum norm error estimate of the discrete solution and its extrapolation. Numerical experiments are presented to support our theoretical results.     

Locally linearized fractional step methods for nonlinear parabolic problems

This work deals with the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. Via the method of lines approach, we first perform the spatial discretization of the original problem by applying a mimetic finite difference scheme. The system of ordinary differential equations arising from that process is then integrated in time with a linearly implicit fractional step method. For that purpose, we locally decompose the discrete nonlinear diffusion operator using suitable Taylor expansions and a domain decomposition splitting technique. The totally discrete scheme considers implicit time integrations for the linear terms while explicitly handling the nonlinear ones. As a result, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled parallelizable linear subsystems. The convergence of the proposed methods is illustrated by numerical experiments.     

Meshless local Petrov-Galerkin (MLPG) approximation to the two dimensional sine-Gordon equation

During the past few years, the idea of using meshless methods for numerical solution of partial differential equations (PDEs) has received much attention throughout the scientific community, and remarkable progress has been achieved on meshless methods. The meshless local Petrov-Galerkin (MLPG) method is one of the “truly meshless” methods since it does not require any background integration cells. The integrations are carried out locally over small sub-domains of regular shapes, such as circles or squares in two dimensions and spheres or cubes in three dimensions. In this paper the MLPG method for numerically solving the non-linear two-dimensional sine-Gordon (SG) equation is developed. A time-stepping method is employed to deal with the time derivative and a simple predictor-corrector scheme is performed to eliminate the non-linearity. A brief discussion is outlined for numerical integrations in the proposed algorithm. Some examples involving line and ring solitons are demonstrated and the conservation of energy in undamped SG equation is investigated. The final numerical results confirm the ability of proposed method to deal with the unsteady non-linear problems in large domains.     

Compact difference methods applied to initial-boundary value problems for mixed systems

Summary. An initial--boundary value problem to a system of nonlinear partial differential equations, which consists of a hyperbolic and a parabolic part, is taken into consideration. The problem is discretised by a compact finite difference method. An approximation of the numerical solution is constructed, at which the difference scheme is linearised. Nonlinear convergence is proved using the stability of the linearised scheme. Finally, a computational experiment for a noncompact scheme is presented. Received May 20, 1995     

Analysis of a symplectic difference scheme for a coupled nonlinear Schrödinger system

In general, proofs of convergence and stability are difficult for symplectic schemes of nonlinear equations. In this paper, a symplectic difference scheme is proposed for an initial-boundary value problem of a coupled nonlinear Schrödinger system. An important lemma and an induction argument are used to prove the unique solvability, convergence and stability of numerical solutions. An iterative algorithm is also proposed for the symplectic scheme and its convergence is proved. Numerical examples show the efficiency of the symplectic scheme and the correction of our numerical analysis.     

Maximum norm error bound of a linearized difference scheme for a coupled nonlinear Schrödinger equations

In this article, a linearized conservative difference scheme for a coupled nonlinear Schrödinger equations is studied. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown that the difference solution unconditionally converges to the exact solution with second order in the maximum norm. Numerical experiments are presented to support the theoretical results.     

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.     

An adaptive numerical method to handle blow-up in a parabolic system

We study numerical approximations to solutions of a system of two nonlinear diffusion equations in a bounded interval, coupled at the boundary in a nonlinear way. In certain cases the system develops a blow-up singularity in finite time. Fixed mesh methods are not well suited to approximate the problem near the singularity. As an alternative to reproduce the behaviour of the continuous solution, we present an adaptive in space procedure. The scheme recovers the conditions for blow-up and non-simultaneous blow-up. It also gives the correct non-simultaneous blow-up rate and set. Moreover, the numerical simultaneous blow-up rates coincide with the continuous ones in the cases when the latter are known. Finally, we present numerical experiments that illustrate the behaviour of the adaptive method.     

Numerical analysis for a conservative difference scheme to solve the Schrödinger-Boussinesq equation

In this paper, we present a finite difference scheme for the solution of an initial-boundary value problem of the Schrödinger-Boussinesq equation. The scheme is fully implicit and conserves two invariable quantities of the system. We investigate the existence of the solution for the scheme, give computational process for the numerical solution and prove convergence of iteration method by which a nonlinear algebra system for unknown Vⁿ⁺¹ is solved. On the basis of a priori estimates for a numerical solution, the uniqueness, convergence and stability for the difference solution is discussed. Numerical experiments verify the accuracy of our method.     

Convergence and stability of the finite difference scheme for nonlinear parabolic systems with time delay

In this article, a class of nonlinear evolution equations – reaction–diffusion equations with time delay – is studied. By combining the domain decomposition technique and the finite difference method, the results for the existence, convergence and the stability of the numerical solution are obtained in the case of subdomain overlap and when the time-space is completely discretized. This revised version was published online in June 2006 with corrections to the Cover Date.     

A convergent scheme for a non local Hamilton Jacobi equation modelling dislocation dynamics

We study dislocation dynamics with a level set point of view. The model we present here looks at the zero level set of the solution of a non local Hamilton Jacobi equation, as a dislocation in a plane of a crystal. The front has a normal speed, depending on the solution itself. We prove existence and uniqueness for short time in the set of continuous viscosity solutions. We also present a first order finite difference scheme for the corresponding level set formulation of the model. The scheme is based on monotone numerical Hamiltonian, proposed by Osher and Sethian. The non local character of the problem makes it not monotone. We obtain an explicit convergence rate of the approximate solution to the viscosity solution. We finally provide numerical simulations.This work has been supported by funds from ACI JC 1041 “Mouvements d’interfaces avec termes non-locaux”, from ACI-JC 1025 “Dynamique des dislocations” and from ONERA, Office National d’Etudes et de Recherches. The second author was also supported by the ENPC-Région Ile de France.     

A third-order semi-discrete genuinely multidimensional central scheme for hyperbolic conservation laws and related problems

Summary. We construct a new third-order semi-discrete genuinely multidimensional central scheme for systems of conservation laws and related convection-diffusion equations. This construction is based on a multidimensional extension of the idea, introduced in [17] – the use of more precise information about the local speeds of propagation, and integration over nonuniform control volumes, which contain Riemann fans. As in the one-dimensional case, the small numerical dissipation, which is independent of , allows us to pass to a limit as . This results in a particularly simple genuinely multidimensional semi-discrete scheme. The high resolution of the proposed scheme is ensured by the new two-dimensional piecewise quadratic non-oscillatory reconstruction. First, we introduce a less dissipative modification of the reconstruction, proposed in [29]. Then, we generalize it for the computation of the two-dimensional numerical fluxes. Our scheme enjoys the main advantage of the Godunov-type central schemes –simplicity, namely it does not employ Riemann solvers and characteristic decomposition. This makes it a universal method, which can be easily implemented to a wide variety of problems. In this paper, the developed scheme is applied to the Euler equations of gas dynamics, a convection-diffusion equation with strongly degenerate diffusion, the incompressible Euler and Navier-Stokes equations. These numerical experiments demonstrate the desired accuracy and high resolution of our scheme. Received February 7, 2000 / Published online December 19, 2000     

A stabilized semi-implicit Galerkin scheme for Navier-Stokes equations

By introducing a time relaxation term for the time derivative of higher frequency components, we proposed a stabilized semi-implicit Galerkin scheme for evolutionary Navier-Stokes equations in this paper. Analysis shows that such a scheme has weaker stability conditions than that of a classical semi-implicit Galerkin scheme and, when a suitable relaxation parameter σ is chosen, it generates an approximate solution with the same accuracy as the classical one. That means the proposed scheme might use a larger time step to generate a bounded approximate solution. Thus it is more suitable for long time simulations.     

An efficient nonlinear iteration scheme for nonlinear parabolic-hyperbolic system

A nonlinear iteration method named the Picard-Newton iteration is studied for a two-dimensional nonlinear coupled parabolic-hyperbolic system. It serves as an efficient method to solve a nonlinear discrete scheme with second spatial and temporal accuracy. The nonlinear iteration scheme is constructed with a linearization-discretization approach through discretizing the linearized systems of the original nonlinear partial differential equations. It can be viewed as an improved Picard iteration, and can accelerate convergence over the standard Picard iteration. Moreover, the discretization with second-order accuracy in both spatial and temporal variants is introduced to get the Picard-Newton iteration scheme. By using the energy estimate and inductive hypothesis reasoning, the difficulties arising from the nonlinearity and the coupling of different equation types are overcome. It follows that the rigorous theoretical analysis on the approximation of the solution of the Picard-Newton iteration scheme to the solution of the original continuous problem is obtained, which is different from the traditional error estimate that usually estimates the error between the solution of the nonlinear discrete scheme and the solution of the original problem. Moreover, such approximation is independent of the iteration number. Numerical experiments verify the theoretical result, and show that the Picard-Newton iteration scheme with second-order spatial and temporal accuracy is more accurate and efficient than that of first-order temporal accuracy.     

Space–time discontinuous Galerkin finite element method for shallow water flows

A space–time discontinuous Galerkin (DG) finite element method is presented for the shallow water equations over varying bottom topography. The method results in nonlinear equations per element, which are solved locally by establishing the element communication with a numerical HLLC flux. To deal with spurious oscillations around discontinuities, we employ a dissipation operator only around discontinuities using Krivodonova's discontinuity detector. The numerical scheme is verified by comparing numerical and exact solutions, and validated against a laboratory experiment involving flow through a contraction. We conclude that the method is second order accurate in both space and time for linear polynomials.     

Runge–Kutta convolution quadrature methods for well-posed equations with memory

Runge–Kutta based convolution quadrature methods for abstract, well-posed, linear, and homogeneous Volterra equations, non necessarily of sectorial type, are developed. A general representation of the numerical solution in terms of the continuous one is given. The error and stability analysis is based on this representation, which, for the particular case of the backward Euler method, also shows that the numerical solution inherits some interesting qualitative properties, such as positivity, of the exact solution. Numerical illustrations are provided.     

Stability and accuracy of time-extrapolated ADI-FDTD methods for solving wave equations

Some recent work on the ADI-FDTD method for solving Maxwell's equations in 3-D have brought out the importance of extrapolation methods for the time stepping of wave equations. Such extrapolation methods have previously been used for the solution of ODEs. The present context (of wave equations) brings up two main questions which have not been addressed previously: (1) when will extrapolation in time of an unconditionally stable scheme for a wave equation again feature unconditional stability, and (2) how will the accuracy and computational efficiency depend on how frequently in time the extrapolations are carried out. We analyze these issues here.     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

One-sided stability and convergence of the Nessyahu–Tadmor scheme

Non-oscillatory schemes are widely used in numerical approximations of nonlinear conservation laws. The Nessyahu–Tadmor (NT) scheme is an example of a second order scheme that is both robust and simple. In this paper, we prove a new stability property of the NT scheme based on the standard minmod reconstruction in the case of a scalar strictly convex conservation law. This property is similar to the One-sided Lipschitz condition for first order schemes. Using this new stability, we derive the convergence of the NT scheme to the exact entropy solution without imposing any nonhomogeneous limitations on the method. We also derive an error estimate for monotone initial data.