On symplectic and multisymplectic schemes for the KdV equation

We examine some symplectic and multisymplectic methods for the notorious Korteweg-de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space-time grids.     

A comparison of high resolution schemes for the two-dimensional linear advection equation

Two-dimensional linear advection solutions of transported fluid temperature is explored by implementing high-resolution, explicit finite difference schemes. An application is given to the case of a spatially uniform, oscillating velocity field specified a priori. A comparison of flux-corrected transport (FCT) methods is made with other total variation diminishing (TVD) schemes for the 2-D linear advection of a 2-D Gaussian initial temperature distribution of various half-widths. Further clipping of the sharply peaked Gaussian in 2-D over that which occurs in 1-D is established for the FCT and upwind TVD schemes.     

New schemes for the coupled nonlinear Schrödinger equation

In this paper, we present three new schemes for the coupled nonlinear Schrödinger equation. The three new schemes are multi-symplectic schemes that preserve the intrinsic geometry property of the equation. The three new schemes are also semi-explicit in the sense that they need not solve linear algebraic equations every time-step, which is usually the most expensive in numerical simulation of partial differential equations. Many numerical experiments on collisions of solitons are presented to show the efficiency of the new multi-symplectic schemes.     

Variable mesh difference schemes for solving a nonlinear Schrödinger equation with a linear damping term

This paper describes moving variable mesh finite difference schemes to numerically solve the nonlinear Schrödinger equation including the effects of damping and nonhomogeneity in the propagation media. These schemes have accurately predicted the location of the peak of soliton compared to the uniform mesh, for the case in which the exact solution is known. Numerical results are presented when damping and nonhomogeneous effects are included, and in the absence of these effects the results were verified with the available exact solution.     

Numerical schemes for the forward–backward heat equation

This paper is concerned with a class of forward–backward heat equations. We use Saulyev's scheme to formulate certain approximation schemes. Then a non-overlap domain decomposition method is presented for the numerical solution. The numerical experiments show that the given algorithm is feasible and effective.     

A new unconditionally stable ADI compact scheme for the two-space-dimensional linear hyperbolic equation

We formulate a new alternating direction implicit compact scheme of O(τ²+h ⁴) for the linear hyperbolic equation u _tt +2α u _t +β² u=u _xx +u _yy +f(x, y, t), 0<x, y<1, 0<t≤T, subject to appropriate initial and Dirichlet boundary conditions, where α>0 and β≥0 are real numbers. In this article, we show the method is unconditionally stable by the Von Neumann method. At last, numerical demonstrations are given to illustrate our result.     

An explicit finite-difference scheme for the solution of the kadomtsev-petviashvili equation

A finite-difference method is used to transform the initial/boundary-value problem associated with the nonlinear Kadomtsev-Petviashvili equation, into an explicit scheme.

The numerical method is developed by replacing the time and space derivatives by central-difference approximants. The resulting finite-difference method is analysed for local truncation error, stability and convergence. The results of a number of numerical experiments are given.     

Note on nonstability of the linear functional equation of higher order

We provide a complete solution of the problem of Hyers-Ulam stability for a large class of higher order linear functional equations in single variable, with constant coefficients. We obtain this by showing that such an equation is nonstable in the case where at least one of the roots of the characteristic equation is of module 1. Our results are related to the notions of shadowing (in dynamical systems and computer science) and controlled chaos. They also correspond to some earlier results on approximate solutions of functional equations in single variable.     

Multilevel difference schemes for the heat conduction equation and its application to the dirichlet problem in two and three dimensions

In this paper we have discussed a general method of deriving high order multilevel schemes for the heat conduction equation in higher dimensions. The application of the three level parabolic schemes for the solution of the steady state Dirichlet problem in two and three dimensions is discussed by introducing two parameters. The error bounds given by Hadjidimos are further refined which drastically reduces the theoretical estimate of the number of iteration cycles required for a given accuracy η. We have also defined a new set of iteration parameters. A numerical example is computed and it is seen that the present methods produce accurate results.     

Application of a fourth-order compact ADI method to solve a two-dimensional linear hyperbolic equation

In this paper, a compact alternating direction implicit method is developed for solving a linear hyperbolic equation with constant coefficients. Its stability criterion is determined by using von Neumann method. It is shown through a discrete energy method that this method can attain fourth-order accuracy in both time and space with respect to H ¹- and L ²-norms provided the stability condition is fulfilled. Its solvability is also analysed in detail. Numerical results confirm the convergence orders and efficiency of our algorithm.     

Numerical study of a nonlinear heat equation for plasma physics

This paper is devoted to the numerical approximation of a nonlinear parabolic balance equation, which describes the heat evolution of a magnetically confined plasma in the edge region of a tokamak. The nonlinearity implies some numerical difficulties, in particular for the long-time behaviour approximation, when solved with standard methods. An efficient numerical scheme is presented in this paper, based on a combination of a directional splitting scheme and the implicit–explicit scheme introduced in Filbet and Jin [A class of asymptotic preserving schemes for kinetic equations and related problems with stiff sources, J. Comput. Phys. 229 (2010), pp. 7625–7648].     

On stable and explicit numerical methods for the advection–diffusion equation

In this paper two stable and explicit numerical methods to integrate the one-dimensional (1D) advection–diffusion equation are presented. These schemes are stable by design and follow the main general concept behind the semi-Lagrangian method by constructing a virtual grid where the explicit method becomes stable. It is shown that the new schemes compare well with analytic solutions and are often more accurate than implicit schemes. In particular, the diffusion-only case is explored in some detail. The error produced by the stable and explicit method is a function of the ratio between the standard deviation σ₀ of the initial Gaussian state and the characteristic virtual grid distance ΔS. Larger values of this ratio lead to very accurate results when compared to implicit methods, while lower values lead to less accuracy. It is shown that the σ₀/ΔS ratio is also significant in the advection–diffusion problem: it determines the maximum error generated by new methods, obtained with a certain combination of the advection and diffusion values. In addition, the error becomes smaller when the problem becomes more advective or more diffusive.     

On Symplectic and Multisymplectic Schemes for the KdV Equation

We examine some symplectic and multisymplectic methods for the notorious Korteweg–de Vries equation, with the question whether the added structure preservation that these methods offer is key in providing high quality schemes for the long time integration of nonlinear, conservative partial differential equations. Concentrating on second order discretizations, several interesting schemes are constructed and studied. Our essential conclusions are that it is possible to design very stable, conservative difference schemes for the nonlinear, conservative KdV equation. Among the best of such schemes are methods which are symplectic or multisymplectic. Semi-explicit, symplectic schemes can be very effective in many situations. Compact box schemes are effective in ensuring that no artificial wiggles appear in the approximate solution. A family of box schemes is constructed, of which the multisymplectic box scheme is a prominent member, which are particularly stable on coarse space–time grids     

Generalized trapezoidal formulas for the black–scholes equation of option pricing

For the celebrated Black–Scholes parabolic equation of option pricing, we present new time integration schemes based on the generalized trapezoidal formulas introduced by Chawla et al. [3]. The resulting GTF(α) schemes are unconditionally stable and second order in both space and time. Interestingly, since the Black–Scholes equation is linear, GTF (1/3) attains order three in time. The computational performance of the obtained schemes is compared with the Crank–Nicolson scheme for the case of European option valuation. Since the payoff is nondifferentiable having a “corner” on expiry at the exercise price, the classical trapezoidal formula used in the Crank–Nicolson scheme can experience oscillations at this corner. It is demonstrated that our present GTF (1/3) scheme can cope with this situation and performs consistently superior than the Crank–Nicolson scheme.     

Compact finite volume methods for the diffusion equation

We describe an approach to treating initial-boundary-value problems by finite volume methods in which the parallel between differential and difference arguments is closely maintained. By using intrinsic geometrical properties of the volume elements, we are able to describe discrete versions of the div, curl, and grad operators which lead, using summation-by-parts techniques, to familiar energy equations as well as the div curl=0 and curl grad=0 identities. For the diffusion equation, these operators describe compact schemes whose convergence is assured by the energy equations and which yield both the potential and the flux vector with second-order accuracy. A simplified potential form is especially useful for obtaining numerical results by multigrid and ADI methods. The treatment of general curvilinear coordinates is shown to result from a specialization of these general results.     

Unconditionally stable ADI scheme of higher-order for linear hyperbolic equations

An unconditionally stable alternating direction implicit (ADI) method of higher-order in space is proposed for solving two- and three-dimensional linear hyperbolic equations. The method is fourth-order in space and second-order in time. The solution procedure consists of a multiple use of one-dimensional matrix solver which produces a computational cost effective solver. Numerical experiments are conducted to compare the new scheme with the existing scheme based on second-order spatial discretization. The effectiveness of the new scheme is exhibited from the numerical results.     

Composition schemes for the stochastic differential equation describing collisional pitch-angle diffusion

Two new second order accurate Monte Carlo integration schemes are derived for the stochastic differential equation describing pitch-angle scattering by Coulomb collisions in magnetized plasmas. Here the pitch-angle is the angle between the magnetic field and the particle velocity vectors. Mathematically this collision process corresponds to diffusion in the polar angle of a spherical coordinate system. The schemes are simple to implement, they are naturally bounded to the solution domain and their convergences are shown to compare favourably against commonly used alternative integration schemes.