On some finite difference schemes for solution of hyperbolic heat conduction problems

We consider the accuracy of two finite difference schemes proposed recently in [Roy S., Vasudeva Murthy A.S., Kudenatti R.B., A numerical method for the hyperbolic-heat conduction equation based on multiple scale technique, Appl. Numer. Math., 2009, 59(6), 1419–1430], and [Mickens R.E., Jordan P.M., A positivity-preserving nonstandard finite difference scheme for the damped wave equation, Numer. Methods Partial Differential Equations, 2004, 20(5), 639–649] to solve an initial-boundary value problem for hyperbolic heat transfer equation. New stability and approximation error estimates are proved and it is noted that some statements given in the above papers should be modified and improved. Finally, two robust finite difference schemes are proposed, that can be used for both, the hyperbolic and parabolic heat transfer equations. Results of numerical experiments are presented.     

New conservative schemes with discrete variational derivatives for nonlinear wave equations

New conservative finite difference schemes for certain classes of nonlinear wave equations are proposed. The key tool there is “discrete variational derivative”, by which discrete conservation property is realized. A similar approach for the target equations was recently proposed by Furihata, but in this paper a different approach is explored, where the target equations are first transformed to the equivalent system representations which are more natural forms to see conservation properties. Applications for the nonlinear Klein–Gordon equation and the so-called “good” Boussinesq equation are presented. Numerical examples reveal the good performance of the new schemes.     

Conservative finite difference methods for fractional Schrödinger–Boussinesq equations and convergence analysis

In this paper, two conservative finite difference schemes for fractional Schrödinger–Boussinesq equations are formulated and investigated. The convergence of the nonlinear fully implicit scheme is established via discrete energy method, while the linear semi‐implicit scheme is analyzed by means of mathematical induction method. Our schemes are proved to preserve the total mass and energy in discrete level. The numerical results are given to confirm the theoretical analysis.     

Discretization of multidimensional submanifolds associated with spin-valued spectral problems

We present a large family of Spin(p, q)-valued discrete spectral problems. The associated discrete nets generated by the so-called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in ℝ^m including isothermic surfaces, Guichard nets, and some other families of orthogonal nets. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 1, pp. 253–262, 2006.     

Malliavin calculus for difference approximations of multidimensional diffusions: Truncated local limit theorem

The truncated local limit theorem is proved for difference approximations of multidimensional diffusions. Under very mild conditions on the distributions of difference terms, this theorem states that the transition probabilities of these approximations, after truncation of some asymptotically negligible terms, possess densities uniformly convergent to the transition probability density for the limiting diffusion and satisfy certain uniform diffusion-type estimates. The proof is based on a new version of the Malliavin calculus for the product of a finite family of measures that may contain nontrivial singular components. Applications to the uniform estimation of mixing and convergence rates for difference approximations of stochastic differential equations and to the convergence of difference approximations of local times for multidimensional diffusions are presented. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 340–381, March, 2008.     

High‐order energy‐preserving schemes for the improved Boussinesq equation

This article proposes a class of high‐order energy‐preserving schemes for the improved Boussinesq equation. To derive the energy‐preserving schemes, we first discretize the improved Boussinesq equation by Fourier pseudospectral method, which leads to a finite‐dimensional Hamiltonian system. Then, the obtained semidiscrete system is solved by Hamiltonian boundary value methods, which is a newly developed class of energy‐preserving methods. The proposed schemes can reach spectral precision in space, and in time can reach second‐order, fourth‐order, and sixth‐order accuracy, respectively. Moreover, the proposed schemes can conserve the discrete mass and energy to within machine precision. Furthermore, to show the efficiency and accuracy of the proposed methods, the proposed methods are compared with the finite difference methods and the finite volume element method. The results of several numerical experiments are given for the propagation of the single solitary wave, the interaction of two solitary waves and the wave break‐up.     

Convergent discretizations for the Nernst–Planck–Poisson system

We propose and compare two classes of convergent finite element based approximations of the nonstationary Nernst–Planck–Poisson equations, whose constructions are motivated from energy versus entropy decay properties for the limiting system. Solutions of both schemes converge to weak solutions of the limiting problem for discretization parameters tending to zero. Our main focus is to study qualitative properties for the different approaches at finite discretization scales, like conservation of mass, non-negativity, discrete maximum principle, decay of discrete energies, and entropies to study long-time asymptotics.     

Numerical simulations of the improved Boussinesq equation

In this study, numerical simulations of the improved Boussinesq equation are obtained using two finite difference schemes and two finite element methods, based on the second‐and third‐order time discretization. The methods are tested on the problems of propagation of a soliton and interaction of two solitons. After the L_∞ error norm is used to measure differences between the exact and numerical solutions, the results obtained by the proposed methods are compared with recently published results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A finite difference analysis of a streamline diffusion method on a Shishkin mesh

We consider streamline diffusion finite element methods applied to a singularly perturbed convection–diffusion two‐point boundary value problem whose solution has a single boundary layer. To analyse the convergence of these methods, we rewrite them as finite difference schemes. We first consider arbitrary meshes, then, in analysing the scheme on a Shishkin mesh, we consider two formulations on the fine part of the mesh: the usual streamline diffusion upwinding and the standard Galerkin method. The error estimates are given in the discrete L _∞ norm; in particular we give the first analysis that shows precisely how the error depends on the user-chosen parameter _τ0 specifying the mesh. When _τ0 is too small, the error becomes O(1), but for _τ0 above a certain threshold value, the error is small and increases either linearly or quadratically as a function of . Numerical tests support our theoretical results. This revised version was published online in August 2006 with corrections to the Cover Date.     

Compact Difference Schemes for the Multidimensional Klein–Gordon Equation

Differential Equations - Stable compact difference schemes of $$4+2$$ and $$4+4 $$ approximation orders are considered and studied on standard stencils for the multidimensional Klein–Gordon...     

On well‐posedness of nonclassical problems for elliptic equations

In the present paper, we consider nonclassical problems for multidimensional elliptic equations. A finite difference method for solving these nonlocal boundary value problems is presented. Stability, almost coercive stability and coercive stability for the solutions of first and second orders of approximation are obtained. The theoretical statements for the solutions of these difference schemes are supported by numerical examples for the two‐dimensional elliptic equations. Copyright © 2014 John Wiley & Sons, Ltd.     

Family of finite-difference schemes with approximate transparent boundary conditions for the generalized nonstationary Schrödinger equation in a semi-infinite strip

An initial-boundary value problem for the generalized Schrödinger equation in a semi-infinite strip is solved. A new family of two-level finite-difference schemes with averaging over spatial variables on a finite mesh is constructed, which covers a set of finite-difference schemes built using various methods. For the family, an abstract approximate transparent boundary condition (TBC) is formulated and the solutions are proved to be absolutely stable in two norms with respect to both initial data and free terms. A discrete TBC is derived, and the stability of the family of schemes with this TBC is proved. The implementation of schemes with the discrete TBC is discussed, and numerical results are presented.     

A second order numerical scheme for the solution of the one-dimensional Boussinesq equation

A predictor–corrector (P-C) scheme is applied successfully to a nonlinear method arising from the use of rational approximants to the matrix-exponential term in a three-time level recurrence relation. The resulting nonlinear finite-difference scheme, which is analyzed for local truncation error and stability, is solved using a P-C scheme, in which the predictor and the corrector are explicit schemes of order 2. This scheme is accelerated by using a modification (MPC) in which the already evaluated values are used for the corrector. The behaviour of the P-C/MPC schemes is tested numerically on the Boussinesq equation already known from the bibliography free of boundary conditions. The numerical results are derived for both the bad and the good Boussinesq equation and conclusions from the relevant known results are derived.      

Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).     

A numerical study of variable depth KdV equations and generalizations of Camassa–Holm-like equations

In this paper we numerically study the KdV-top equation and compare it with the Boussinesq equations over uneven bottoms. We use here a finite-difference scheme that conserves a discrete energy for the fully discrete scheme. We also compare this approach with the discontinuous Galerkin method. For the equations obtained in the case of stronger nonlinearities and related to the Camassa–Holm equation, we find several finite difference schemes that conserve a discrete energy for the fully discrete scheme. Because of its accuracy for the conservation of energy, our numerical scheme is also of interest even in the simple case of flat bottoms. We compare this approach with the discontinuous Galerkin method.     

A finite difference approach for the initial-boundary value problem of the fractional Klein-Kramers equation in phase space

Considering the features of the fractional Klein-Kramers equation (FKKE) in phase space, only the unilateral boundary condition in position direction is needed, which is different from the bilateral boundary conditions in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. In the paper, a finite difference scheme is constructed, where temporal fractional derivatives are approximated using L1 discretization. The advantages of the scheme are: for every temporal level it can be dealt with from one side to the other one in position direction, and for any fixed position only a tri-diagonal system of linear algebraic equations needs to be solved. The computational amount reduces compared with the ADI scheme in [Cartling B., Kinetics of activated processes from nonstationary solutions of the Fokker-Planck equation for a bistable potential, J. Chem. Phys., 1987, 87(5), 2638–2648] and the five-point scheme in [Deng W., Li C., Finite difference methods and their physical constrains for the fractional Klein-Kramers equation, Numer. Methods Partial Differential Equations, 2011, 27(6), 1561–1583]. The stability and convergence are proved and two examples are included to show the accuracy and effectiveness of the method.     

Approximation of optimal feedback control: a dynamic programming approach

We consider the general continuous time finite-dimensional deterministic system under a finite horizon cost functional. Our aim is to calculate approximate solutions to the optimal feedback control. First we apply the dynamic programming principle to obtain the evolutive Hamilton–Jacobi–Bellman (HJB) equation satisfied by the value function of the optimal control problem. We then propose two schemes to solve the equation numerically. One is in terms of the time difference approximation and the other the time-space approximation. For each scheme, we prove that (a) the algorithm is convergent, that is, the solution of the discrete scheme converges to the viscosity solution of the HJB equation, and (b) the optimal control of the discrete system determined by the corresponding dynamic programming is a minimizing sequence of the optimal feedback control of the continuous counterpart. An example is presented for the time-space algorithm; the results illustrate that the scheme is effective.     

Convergence analysis of some high–order time accurate schemes for a finite volume method for second order hyperbolic equations on general nonconforming multidimensional spatial meshes

The present work is an extension of our previous work (Bradji, Numer Methods Partial Differ Equations, to appear) which dealt with error analysis of a finite volume scheme of a first convergence order (both in time and space) for second‐order hyperbolic equations on general nonconforming multidimensional spatial meshes introduced recently in (Eymard et al. IMAJ Numer Anal 30(2010), 1009–1043). We aim in this article to get some higher‐order time accurate schemes for a finite volume method for second‐order hyperbolic equations using the same class of spatial generic meshes stated above. We derive a family of finite volume schemes approximating the wave equation, as a model for second‐order hyperbolic equations, in which the discretization in time is performed using a one‐parameter scheme of the Newmark's method. We prove that the error estimate of these finite volume schemes is of order two (or four) in time and it is of optimal order in space. These error estimates are analyzed in several norms which allow us to derive approximations for the exact solution and its first derivatives whose the convergence order is two (or four) in time and it is optimal in space. We prove in particular, when the discrete flux is calculated using a stabilized discrete gradient, that the convergence order is \begin{align*}k^2+h_\mathcal{D}\end{align*} or \begin{align*}k^4+h_\mathcal{D}\end{align*}, where \begin{align*}h_\mathcal{D}\end{align*} (resp. k) is the mesh size of the spatial (resp. time) discretization. These estimates are valid under the regularity assumption \begin{align*}u\in C^4(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are second‐order accurate in time, and \begin{align*}u\in C^6(\lbrack 0,T\rbrack;C^2(\overline{\Omega}))\end{align*}, when the schemes are four‐order accurate in time for the exact solution u. The proof of these error estimates is based essentially on a comparison between the finite volume approximate solution and an auxiliary finite volume approximation. © 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Quasi-least-squares finite element method for steady flow and heat transfer with system rotation

Two quasi-least-squares finite element schemes based on L ² inner product are proposed to solve a steady Navier–Stokes equations, coupled to the energy equation by the Boussinesq approximation and augmented by a Coriolis forcing term to account for system rotation. The resulting nonlinear systems are linearized around a characteristic state, resulting in linearized least-squares models that yield algebraic systems with symmetric positive definite coefficient matrices. Existence of solutions are investigated and a priori error estimates are obtained. The performance of the formulation is illustrated by using a direct iteration procedure to treat the nonlinearities and shown theoretical convergent rate for general initial guess.     

$L^1$-Stability and error estimates for approximate Hamilton-Jacobi solutions

Summary.   We study the -stability and error estimates of general approximate solutions for the Cauchy problem associated with multidimensional Hamilton-Jacobi (H-J) equations. For strictly convex Hamiltonians, we obtain a priori error estimates in terms of the truncation errors and the initial perturbation errors. We then demonstrate this general theory for two types of approximations: approximate solutions constructed by the vanishing viscosity method, and by Godunov-type finite difference methods. If we let denote the `small scale' of such approximations (– the viscosity amplitude , the spatial grad-size , etc.), then our -error estimates are of , and are sharper than the classical -results of order one half, . The main building blocks of our theory are the notions of the semi-concave stability condition and -measure of the truncation error. The whole theory could be viewed as a multidimensional extension of the -stability theory for one-dimensional nonlinear conservation laws developed by Tadmor et. al. [34,24,25]. In addition, we construct new Godunov-type schemes for H-J equations which consist of an exact evolution operator and a global projection operator. Here, we restrict our attention to linear projection operators (first-order schemes). We note, however, that our convergence theory applies equally well to nonlinear projections used in the context of modern high-resolution conservation laws. We prove semi-concave stability and obtain -bounds on their associated truncation errors; -convergence of order one then follows. Second-order (central) Godunov-type schemes are also constructed. Numerical experiments are performed; errors and orders are calculated to confirm our -theory. Received April 20, 1998 / Revised version received November 8, 1999 / Published online August 24, 2000