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 collocation method with cubic B-splines for solving the MRLW equation

The modified regularized long wave (MRLW) equation is solved numerically by collocation method using cubic B-splines finite element. A linear stability analysis of the scheme is shown to be marginally stable. Three invariants of motion are evaluated to determine the conservation properties of the algorithm, also the numerical scheme leads to accurate and efficient results. Moreover, interaction of two and three solitary waves are studied through computer simulation and the development of the Maxwellian initial condition into solitary waves is also shown.     

Construction of solutions to the two-dimensional stationary Euler equations by the pseudo-advection method

The finite difference approximation of a nonstationary pseudo-advected vorticity equation is proved to yield generalized solutions to the two-dimensional stationary Euler equations with nonvanishing vorticity. This result is obtained by the simultaneous limiting of lattice scale and time.Received: 15 May 2002     

A reduced finite difference scheme based on singular value decomposition and proper orthogonal decomposition for Burgers equation

In this article, a reduced optimizing finite difference scheme (FDS) based on singular value decomposition (SVD) and proper orthogonal decomposition (POD) for Burgers equation is presented. Also the error estimates between the usual finite difference solution and the POD solution of reduced optimizing FDS are analyzed. It is shown by considering the results obtained for numerical simulations of cavity flows that the error between the POD solution of reduced optimizing FDS and the solution of the usual FDS is consistent with theoretical results. Moreover, it is also shown that the reduced optimizing FDS is feasible and efficient.     

A penalty finite element method based on the Euler implicit/explicit scheme for the time-dependent Navier-Stokes equations

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations, where the time discretization is based on the Euler implicit/explicit scheme with some implicit linear terms and an explicit nonlinear term, and the finite element spatial discretization is based on the P₁b-P₁ element pair, which satisfies the discrete inf-sup condition. This method allows us to separate the computation of the velocity from the computation of the pressure with a larger time-step size Δt, so that the numerical velocity and the pressure are easily computed. An optimal error estimate of the numerical velocity and the pressure is provided for the fully discrete penalty finite element method when the penalty parameter ?, the time-step size Δt and the mesh size h satisfy the following stability conditions: ?c₁≤1, Δtκ₁≤1 and h²≤β₁Δt, respectively, for some positive constants c₁, κ₁ and β₁. Finally, some numerical tests to confirm the theoretical results of the penalty finite element method are provided.     

Overlapping Schwarz waveform relaxation method for the solution of the forward–backward heat equation

In this article we analyzed the convergence of the Schwarz waveform relaxation method for solving the forward–backward heat equation. Numerical results are presented for a specific type of model problem.     

High order compact Alternating Direction Implicit method for the generalized sine-Gordon equation

High order compact Alternating Direction Implicit scheme is given for solving the generalized sine-Gordon equation in a two-dimensional rectangular domain. We apply the compact finite difference operators to obtain a fourth order discretization for the second order space derivatives, and we give a linearized three time level algorithm for solving the original nonlinear equation. Error estimate is given by the energy method. Numerical results are provided to verify the accuracy and efficiency of this algorithm.     

Numerical schemes for a pseudo-parabolic Burgers equation: Discontinuous data and long-time behaviour

We consider a simplified model for vertical non-stationary groundwater flow, which includes dynamic capillary pressure effects. Specifically, we consider a viscous Burgers-type equation that is extended with a third-order term containing mixed derivatives in space and time. We analyse the one-dimensional boundary value problem and investigate numerically its long-time behaviour. The numerical schemes discussed here take into account possible discontinuities of the solution.     

Global conservative solutions for a model equation for shallow water waves of moderate amplitude

In this paper, we study the continuation of solutions to an equation for surface water waves of moderate amplitude in the shallow water regime beyond wave breaking (in [11], Constantin and Lannes proved that this equation accommodates wave breaking phenomena). Our approach is based on a method proposed by Bressan and Constantin [2]. By introducing a new set of independent and dependent variables, which resolve all singularities due to possible wave breaking, the evolution problem is rewritten as a semilinear system. Local existence of the semilinear system is obtained as fixed points of a contractive transformation. Moreover, this formulation allows one to continue the solution after collision time, giving a global conservative solution where the energy is conserved for almost all times. Finally, returning to the original variables, we obtain a semigroup of global conservative solutions, which depend continuously on the initial data.     

The solution of the two-dimensional sine-Gordon equation using the method of lines

The method of lines is used to transform the initial/boundary-value problem associated with the two-dimensional sine-Gordon equation in two space variables into a second-order initial-value problem. The finite-difference methods are developed by replacing the matrix-exponential term in a recurrence relation with rational approximants. The resulting finite-difference methods are analyzed for local truncation error, stability and convergence. To avoid solving the nonlinear system a predictor–corrector scheme using the explicit method as predictor and the implicit as corrector is applied. Numerical solutions for cases involving the most known from the bibliography line and ring solitons are given.     

A Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation

A new Hamiltonian-conserving Galerkin scheme for the Camassa-Holm equation is presented. The scheme has an additional welcome feature that in exact arithmetic it is unconditionally stable in the sense that the solution is always bounded. Numerical examples that confirm the theory and the effectiveness of the scheme are also given.     

A numerical scheme for periodic travelling-wave simulations in some nonlinear dispersive wave models

A numerical method for simulating periodic travelling-wave solutions of some nonlinear dispersive wave equations is proposed. The construction of the scheme is based on an efficient computation of the elements that characterize these solutions: the initial profile and the velocity of the wave.     

A self-adaptive moving mesh method for the Camassa-Holm equation

A self-adaptive moving mesh method is proposed for the numerical simulations of the Camassa-Holm equation. It is an integrable scheme in the sense that it possesses the exact N-soliton solution. It is named a self-adaptive moving mesh method, because the non-uniform mesh is driven and adapted automatically by the solution. Once the non-uniform mesh is evolved, the solution is determined by solving a tridiagonal linear system. Due to these two superior features of the method, several test problems give very satisfactory results even if by using a small number of grid points.     

An adaptive multigrid scheme for Bose-Einstein condensates in a periodic potential

We present a novel multigrid-continuation method for treating parameter-dependent problems. The proposed algorithm which can be flexibly implemented is a generalization of the two-grid discretization schemes [C.-S. Chien, B.-W. Jeng, A two-grid discretization scheme for semilinear elliptic eigenvalue problems, SIAM J. Sci. Comput. 27 (2006) 1287-1304]. That is, approximating points on a solution curve do not necessarily lie on the same fine grid. We apply the algorithm to compute energy levels and superfluid densities of Bose-Einstein condensates (BEC) in a periodic potential. Both positive and negative scattering lengths are considered in our numerical experiments. For positive scattering length, if the chemical potential is large enough, and the domain is properly chosen, the results show that the number of peaks of the first few energy states of the 2D BEC in a periodic potential depends on the wave number of the periodic potential. Moreover, for bright solitons the number of peaks of the ground state solutions is and , where the periodic potential is expressed in terms of the sine or the cosine functions, respectively. However, these formulae do not hold if the scattering length is negative. The numerical study is extended to the two-component, 1D and 2D BEC in a periodic potential.     

On local convergence of a symmetric semi-discrete scheme for an abstract analogue of the Kirchhoff equation

The present work considers a nonlinear abstract hyperbolic equation with a self-adjoint positive definite operator, which represents a generalization of the Kirchhoff string equation. A symmetric three-layer semi-discrete scheme is constructed for an approximate solution of a Cauchy problem for this equation. Value of the gradient in the nonlinear term of the scheme is taken at the middle point. It makes possible to find an approximate solution at each time step by inverting the linear operator. Local convergence of the constructed scheme is proved. Numerical calculations for different model problems are carried out using this scheme.     

Global conservative solutions of the generalized hyperelastic-rod wave equation

We prove existence of global and conservative solutions of the Cauchy problem for the nonlinear partial differential equation where f is strictly convex or concave and g is locally uniformly Lipschitz. This includes the Camassa-Holm equation (f(u)=u²/2 and g(u)=κu+u²) as well as the hyperelastic-rod wave equation (f(u)=γu²/2 and g(u)=(3−γ)u²/2) as special cases. It is shown that the problem is well-posed for initial data in H¹(R) if one includes a Radon measure that corresponds to the energy of the system with the initial data. The solution is energy preserving. Stability is proved both with respect to initial data and the functions f and g. The proof uses an equivalent reformulation of the equation in terms of Lagrangian coordinates.     

The uniform saturation property for a singularly perturbed reaction-diffusion equation

Although adaptive finite element methods for solving elliptic problems often work well in practice, they are usually not proven to converge. For Poisson like problems, an exception is given by the method of Dörfler ([8]), that was later improved by Morin, Nochetto and Siebert ([11]). In this paper we extend these methods by constructing an adaptive finite element method for a singularly perturbed reaction-diffusion equation that, in energy norm, converges uniformly in the size of the reaction term. Moreover, in this algorithm the arising Galerkin systems are solved only inexactly, so that, generally, the number of arithmetic operations is equivalent to the number of triangles in the final partition.This work was supported by the Netherlands Organization for Scientific Research and by the EU-IHP project “Breaking Complexity.”     

On the rate of convergence of the binomial tree scheme for American options

An American put option can be modelled as a variational inequality. With a penalization approximation to this variational inequality, the convergence rate of the Binomial Tree Scheme is obtained in this paper.     

A forward scheme for backward SDEs

We introduce a forward scheme for simulating backward SDEs. Compared to existing schemes, ours avoids high order nestings of conditional expectations backwards in time. In this way the error, when approximating the conditional expectation, depending on the time partition, is significantly reduced. Besides this generic result, we present an implementable algorithm and prove its convergence. Finally, we demonstrate the strength of the new algorithm by solving a financial problem numerically.     

A time- and spaceadaptive algorithm for the linear time-dependent Schrödinger equation

Summary. We prove an a posteriori error estimate for the linear time-dependent Schr?dinger equation in . From this, we derive a residual based local error estimator that allows us to adjust the mesh and the time step size in order to obtain a numerical solution with a prescribed accuracy. As a special feature, the error estimator controls localization and size of the finite computational domain in each time step. An algorithm is described to compute this solution and numerical results in one space dimension are included. Received March 17, 1995