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.     

Weak second order S-ROCK methods for Stratonovich stochastic differential equations

It is well known that the numerical solution of stiff stochastic ordinary differential equations leads to a step size reduction when explicit methods are used. This has led to a plethora of implicit or semi-implicit methods with a wide variety of stability properties. However, for stiff stochastic problems in which the eigenvalues of a drift term lie near the negative real axis, such as those arising from stochastic partial differential equations, explicit methods with extended stability regions can be very effective. In the present paper our aim is to derive explicit Runge–Kutta schemes for non-commutative Stratonovich stochastic differential equations, which are of weak order two and which have large stability regions. This will be achieved by the use of a technique in Chebyshev methods for ordinary differential equations.     

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.     

Weighted finite difference methods for a class of space fractional partial differential equations with variable coefficients

A class of weighted finite difference methods (WFDMs) for solving a class of initial-boundary value problems of space fractional partial differential equations with variable coefficients is presented. Their stability and convergence properties are considered. It is proven that the WFDMs are unconditionally-stable for , and conditionally-stable for , here r is the weighting parameter subjected to 0≤r≤1. Some convergence results are given. These methods, problems and results generalize the corresponding methods, problems and results given in [7], [8] and [10]. Some numerical examples are provided to show the effectiveness of the methods with different weighting parameters.     

A fictitious domain approach to the numerical solution of PDEs in stochastic domains

We present an efficient method for the numerical realization of elliptic PDEs in domains depending on random variables. Domains are bounded, and have finite fluctuations. The key feature is the combination of a fictitious domain approach and a polynomial chaos expansion. The PDE is solved in a larger, fixed domain (the fictitious domain), with the original boundary condition enforced via a Lagrange multiplier acting on a random manifold inside the new domain. A (generalized) Wiener expansion is invoked to convert such a stochastic problem into a deterministic one, depending on an extra set of real variables (the stochastic variables). Discretization is accomplished by standard mixed finite elements in the physical variables and a Galerkin projection method with numerical integration (which coincides with a collocation scheme) in the stochastic variables. A stability and convergence analysis of the method, as well as numerical results, are provided. The convergence is “spectral” in the polynomial chaos order, in any subdomain which does not contain the random boundaries.     

Approximation of stochastic advection diffusion equations with Stochastic Alternating Direction Explicit methods

The numerical solutions of stochastic partial differential equations of Itô type with time white noise process, using stable stochastic explicit finite difference methods are considered in the paper. Basically, Stochastic Alternating Direction Explicit (SADE) finite difference schemes for solving stochastic time dependent advection-diffusion and diffusion equations are represented and the main properties of these stochastic numerical methods, e.g. stability, consistency and convergence are analyzed. In particular, it is proved that when stable alternating direction explicit schemes for solving linear parabolic PDEs are developed to the stochastic case, they retain their unconditional stability properties applying to stochastic advection-diffusion and diffusion SPDEs. Numerically, unconditional stable SADE techniques are significant for approximating the solutions of the proposed SPDEs because they do not impose any restrictions for refining the computational domains. The performance of the proposed methods is tested for stochastic diffusion and advection-diffusion problems, and the accuracy and efficiency of the numerical methods are demonstrated.     

Error and extrapolation of a compact LOD method for parabolic differential equations

This paper is concerned with a compact locally one-dimensional (LOD) finite difference method for solving two-dimensional nonhomogeneous parabolic differential equations. An explicit error estimate for the finite difference solution is given in the discrete infinity norm. It is shown that the method has the accuracy of the second-order in time and the fourth-order in space with respect to the discrete infinity norm. A Richardson extrapolation algorithm is developed to make the final computed solution fourth-order accurate in both time and space when the time step equals the spatial mesh size. Numerical results demonstrate the accuracy and the high efficiency of the extrapolation algorithm.     

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.     

A convergence analysis of hopscotch methods for fourth order parabolic equations

Summary Consider the ODE (ordinary differential equation) that arises from a semi-discretization (discretization of the spatial coordinates) of a first order system form of a fourth order parabolic PDE (partial differential equation). We analyse the stability of the finite difference methods for this fourth order parabolic PDE that arise if one applies the hopscotch idea to this ODE.Often the error propagation of these methods can be represented by a three terms matrix-vector recursion in which the matrices have a certain anti-hermitian structure. We find a (uniform) expression for the stability bound (or error propagation bound) of this recursion in terms of the norms of the matrices. This result yields conditions under which these methods are strongly asymptotically stable (i.e. the stability is uniform both with respect to the spatial and the time stepsizes (tending to 0) and the time level (tending to infinity)), also in case the PDE has (spatial) variable coefficients. A convergence theorem follows immediately.     

High-order compact finite difference scheme for option pricing in stochastic volatility models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.     

Three-stage stochastic Runge–Kutta methods for stochastic differential equations

In this paper we discuss three-stage stochastic Runge–Kutta (SRK) methods with strong order 1.0 for a strong solution of Stratonovich stochastic differential equations (SDEs). Higher deterministic order is considered. Two methods, a three-stage explicit (E3) method and a three-stage semi-implicit (SI3) method, are constructed in this paper. The stability properties and numerical results show the effectiveness of these methods in the pathwise approximation of several standard test problems.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

Solving parabolic and hyperbolic equations by the generalized finite difference method

Classical finite difference schemes are in wide use today for approximately solving partial differential equations of mathematical physics. An evolution of the method of finite differences has been the development of generalized finite difference (GFD) method, that can be applied to irregular grids of points.     

Nonlinear scheme with high accuracy for nonlinear coupled parabolic-hyperbolic system

A nonlinear finite difference scheme with high accuracy is studied for a class of two-dimensional nonlinear coupled parabolic-hyperbolic system. Rigorous theoretical analysis is made for the stability and convergence properties of the scheme, which shows it is unconditionally stable and convergent with second order rate for both spatial and temporal variables. In the argument of theoretical results, difficulties arising from the nonlinearity and coupling between parabolic and hyperbolic equations are overcome, by an ingenious use of the method of energy estimation and inductive hypothesis reasoning. The reasoning method here differs from those used for linear implicit schemes, and can be widely applied to the studies of stability and convergence for a variety of nonlinear schemes for nonlinear PDE problems. Numerical tests verify the results of the theoretical analysis. Particularly it is shown that the scheme is more accurate and faster than a previous two-level nonlinear scheme with first order temporal accuracy.     

Stability of collocation-based Runge-Kutta-Nyström methods

We analyse the attainable order and the stability of Runge-Kutta-Nyström (RKN) methods for special second-order initial-value problems derived by collocation techniques. Like collocation methods for first-order equations the step point order ofs-stage methods can be raised to 2s for alls. The attainable stage order is one higher and equalss+1. However, the stability results derived in this paper show that we have to pay a high price for the increased stage order.These investigations were supported by the University of Amsterdam who provided the third author with a research grant for spending a total of two years in Amsterdam.     

High-order finite element methods for time-fractional partial differential equations

The aim of this paper is to develop high-order methods for solving time-fractional partial differential equations. The proposed high-order method is based on high-order finite element method for space and finite difference method for time. Optimal convergence rate O((Δt)^2−α+N^−r) is proved for the (r−1)th-order finite element method (r≥2).     

A modified accelerated monotone iterative method for finite difference reaction-diffusion-convection equations

This paper is concerned with monotone algorithms for the finite difference solutions of a class of nonlinear reaction-diffusion-convection equations with nonlinear boundary conditions. A modified accelerated monotone iterative method is presented to solve the finite difference systems for both the time-dependent problem and its corresponding steady-state problem. This method leads to a simple and yet efficient linear iterative algorithm. It yields two sequences of iterations that converge monotonically from above and below, respectively, to a unique solution of the system. The monotone property of the iterations gives concurrently improving upper and lower bounds for the solution. It is shown that the rate of convergence for the sum of the two sequences is quadratic. Under an additional requirement, quadratic convergence is attained for one of these two sequences. In contrast with the existing accelerated monotone iterative methods, our new method avoids computing local maxima in the construction of these sequences. An application using a model problem gives numerical results that illustrate the effectiveness of the proposed method.     

On composite mesh difference methods for hyperbolic differential equations

Summary Difference schemes on more than one mesh, called composite mesh difference methods (CMDM), are considered for hyperbolic equations. A stability proof for a one-dimensional CMDM is presented and a numerical experiment by a CMDM for the inviscid shallow-water equations is described.Research supported by the Swedish Natural Science Research Council (Nfr. 2711-18)     

A new high accuracy locally one-dimensional scheme for the wave equation

In this paper, a new locally one-dimensional (LOD) scheme with error of O(Δt⁴+h⁴) for the two-dimensional wave equation is presented. The new scheme is four layer in time and three layer in space. One main advantage of the new method is that only tridiagonal systems of linear algebraic equations have to be solved at each time step. The stability and dispersion analysis of the new scheme are given. The computations of the initial and boundary conditions for the two intermediate time layers are explicitly constructed, which makes the scheme suitable for performing practical simulation in wave propagation modeling. Furthermore, a comparison of our new scheme and the traditional finite difference scheme is given, which shows the superiority of our new method.     

A singular perturbation of the heat equation with memory

In this paper we consider a hyperbolic equation, with a memory term in time, which can be seen as a singular perturbation of the heat equation with memory. The qualitative properties of the solutions of the initial boundary value problems associated with both equations are studied. We propose numerical methods for the hyperbolic and parabolic models and their stability properties are analyzed. Finally, we include numerical experiments illustrating the performance of those methods.