High-order accurate implicit running schemes

An approach to the construction of high-order accurate implicit predictor-corrector schemes is proposed. The accuracy is improved by choosing a special time integration step for computing numerical fluxes through cell interfaces by using an unconditionally stable implicit scheme. For smooth solutions of advection equations with constant coefficients, the scheme is second-order accurate. Implicit difference schemes for multidimensional advection equations are constructed on the basis of Godunov’s method with splitting over spatial variables as applied to the computation of “large” values at an intermediate layer. The numerical solutions obtained for advection equations and the radiative transfer equations in a vacuum are compared with their exact solutions. The comparison results confirm that the approach is efficient and that the accuracy of the implicit predictor-corrector schemes is improved.     

First-Order and Second-Order,Chaos-Free,Finite Difference Schemes for Fisher Equation

A new class of finite difference schemes is constructed for Fisher partial differential equation i.e. the reaction-diffusion equation with stiff source term: $au(1-u)$. These schemes have the properties that they reduce to high fidelity algorithms in the diffusion-free case namely in which the numerical solutions preserve the properties in the exact solutions for arbitrary time step-size and reaction coefficient α＞0 and all nonphysical spurious solutions including bifurcations and chaos that normally appear in the standard discrete models of Fisher partial differential equation will not occur. The implicit schemes so developed obtain the numerical solutions by solving a single linear algebraic system at each step. The boundness and asymptotic behaviour of numerical solutions obtained by all these schemes are given. The approach constructing the above schemes can be extended to reaction-diffusion equations with other stiff source terms.     

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. AMS subject classification (2000) 65M30, 65M15, 35K60     

Spectral problems with mixed Dirichlet–Neumann boundary conditions: Isospectrality and beyond

In this study, an implicit semi-discrete higher order compact (HOC) scheme, with an averaged time discretization, has been presented for the numerical solution of unsteady two-dimensional (2D) Schrödinger equation. The scheme is second order accurate in time and fourth order accurate in space. The results of numerical experiments are presented, and are compared with analytical solutions and well established numerical results of some other finite difference schemes. In all cases, the present scheme produces highly accurate results with much better computational efficiency.     

Implicit Difference Methods for Degenerate Hyperbolic Equations of Second Order

This paper is a sequel to [2]. A two parameter family of explicit and implicit schemes is constructed for the numerical solution of the degenerate hyperbolic equations of second order. We prove the existence and the uniqueness of the solutions of these schemes. Furthermore, we prove that these schemes are stable for the initial values and that the numerical solution is convergent to the unique generalized solution of the partial differential equation.     

A frictionless contact problem for elastic–viscoplastic materials with normal compliance: Numerical analysis and computational experiments

Summary. In this paper we consider a frictionless contact problem between an elastic–viscoplastic body and an obstacle. The process is assumed to be quasistatic and the contact is modeled with normal compliance. We present a variational formulation of the problem and prove the existence and uniqueness of the weak solution, using strongly monotone operators arguments and Banach's fixed point theorem. We also study the numerical approach to the problem using spatially semi-discrete and fully discrete finite elements schemes with implicit and explicit discretization in time. We show the existence of the unique solution for each of the schemes and derive error estimates on the approximate solutions. Finally, we present some numerical results involving examples in one, two and three dimensions. Received May 20, 2000 / Revised version received January 8, 2001 / Published online June 7, 2001     

Constraint Preserving,Inexact Solution of Implicit Discretizations of Landau–Lifshitz–Gilbert Equations and Consequences for Convergence

The Landau-Lifshitz-Gilbert equation describes dynamics of ferromagnetism. Nonlinearity of the equation and a non-convex side constraint make it difficult to design reliable approximation schemes. In this paper, we discuss the numerical solution of nonlinear systems of equations resulting from implicit, unconditionally convergent discretizations of the problem. Numerical experiments indicate that finite-time blow-up of weak solutions can occur and thereby underline the necessity of the design of reliable discretization schemes that approximate weak solutions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Numerical results for the Klein‐Gordon equation in de Sitter spacetime

We consider the initial value problem for the Klein‐Gordon equation in de Sitter spacetime. We use the central difference scheme on the temporal discretization. We also discretize the spatial variable using the finite element method with implicit and the Crank‐Nicolson schemes for the numerical solution of the initial value problem. In order to show the accuracy for the results of the solutions, we also examine the finite difference methods. We observe that the numerical results obtained by using these methods are compatible.     

Novel numerical techniques for the finite moment log stable computational model for European call option

Option pricing models are often used to describe the dynamic characteristics of prices in financial markets. Unlike the classical Black–Scholes (BS) model, the finite moment log stable (FMLS) model can explain large movements of prices during small time steps. In the FMLS, the second-order spatial derivative of the BS model is replaced by a fractional operator of order α which generates an α-stable Lévy process. In this paper, we consider the finite difference method to approximate the FMLS model. We present two numerical schemes for this approximation: the implicit numerical scheme and the Crank–Nicolson scheme. We carry out convergence and stability analyses for the proposed schemes. Since the fractional operator routinely generates dense matrices which often require high computational cost and storage memory, we explore three methods for solving the approximation schemes: the Gaussian elimination method, the bi-conjugate gradient stabilized method (Bi-CGSTAB) and the fast Bi-CGSTAB (FBi-CGSTAB) in order to compare the cost of calculations. Finally, two numerical examples with exact solutions are presented where we also use extrapolation techniques to achieve higher-order convergence. The results suggest that the proposed schemes are unconditionally stable and convergent, and the FMLS model is useful for pricing options.     

High-order time integration methods in molecular dynamics

The mechanical behaviour of molecular structures can be described with stiff differential equations, which can not be solved analytically. Several numerical time integration schemes can be found in the literature. The aim of this paper is to present the class of partitioned Runge-Kutta methods applied in molecular dynamics. This class of methods includes a wide range of explicit and implicit, single- and multi-stage, symplectic and non-symplectic, low- and high-order time integration schemes. Also most of the classical methods like explicit and implicit Euler, explicit and implicit midpoint rule, Störmer-Verlet and Newmark are also partitioned Runge-Kutta methods. The schemes are implemented in a finite element code which can serve as a numerical platform for molecular dynamics. This code is used to show the sensitivity of the simulations to the accuracy of the initial values. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A numerical method for solving partial differential algebraic equations

Linear systems of partial differential equations with constant coefficient matrices are considered. The matrices multiplying the derivatives of the sought vector function are assumed to be singular. The structure of solutions to such systems is examined. The numerical solution of initialboundary value problems for such equations by applying implicit difference schemes is discussed.     

Nonstandard finite difference schemes for a class of generalized convection–diffusion–reaction equations

In this work, a class of nonstandard finite difference (NSFD) schemes are proposed to approximate the solutions of a class of generalized convection–diffusion–reaction equations. First, in the case of no diffusion, two exact finite difference schemes are presented using the method of characteristics. Based on these two exact schemes, a class of exact schemes are presented by introducing a parameter α. Second, since the forms of these exact schemes are so complicated that they are not convenient to use, a class of NSFD schemes are derived from the exact schemes using numerical approximations. It follows that, under certain conditions about denominator function of time‐step sizes, these NSFD schemes are elementary stable and the solutions are positive and bounded. Third, by means of the Mickens' technique of subequations, a new class of implicit NSFD schemes are constructed for the full convection–diffusion–reaction equations. It is shown that, under certain parameters set, these NSFD schemes are capable of preserving the non‐negativity and boundedness of the analytical solutions. Finally, some numerical simulations are provided to verify the validity of our analytical results. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1288–1309, 2015     

High order accurate semi-implicit WENO schemes for hyperbolic balance laws

In this paper we propose a family of well-balanced semi-implicit numerical schemes for hyperbolic conservation and balance laws. The basic idea of the proposed schemes lies in the combination of the finite volume WENO discretization with Roe’s solver and the strong stability preserving (SSP) time integration methods, which ensure the stability properties of the considered schemes [S. Gottlieb, C.-W. Shu, E. Tadmor, Strong stability-preserving high-order time discretization methods, SIAM Rev. 43 (2001) 89-112]. While standard WENO schemes typically use explicit time integration methods, in this paper we are combining WENO spatial discretization with optimal SSP singly diagonally implicit (SDIRK) methods developed in [L. Ferracina, M.N. Spijker, Strong stability of singly diagonally implicit Runge-Kutta methods, Appl. Numer. Math. 58 (2008) 1675-1686]. In this way the implicit WENO numerical schemes are obtained. In order to reduce the computational effort, the implicit part of the numerical scheme is linearized in time by taking into account the complete WENO reconstruction procedure. With the proposed linearization the new semi-implicit finite volume WENO schemes are designed.A detailed numerical investigation of the proposed numerical schemes is presented in the paper. More precisely, schemes are tested on one-dimensional linear scalar equation and on non-linear conservation law systems. Furthermore, well-balanced semi-implicit WENO schemes for balance laws with geometrical source terms are defined. Such schemes are then applied to the open channel flow equations. We prove that the defined numerical schemes maintain steady state solution of still water. The application of the new schemes to different open channel flow examples is shown.     

Numerical solving dynamic problems of elastoplastic deformation of solids

Numerical schemes for solving two-dimensional dynamic problems of elasticity theory based upon several local approximations for each of the required functions are discussed. The schemes contain free parameters (dissipation constants). An explicit form of artificial dissipation of the solutions allows us to control its size and to effectively construct both explicit and implicit schemes. The principle of producing such schemes is applied to a plane dynamic problem of elasticity theory as an example. We describe a class of problems for which numerical algorithms using several local approximations for each of the required functions are constructed. Examples of solving practical problems are given.     

A Stable Numerical Approach for Implicit Non-Linear Neutral Delay Differential Equations

In this paper we consider implicit non-linear neutral delay differential equations to derive efficient numerical schemes with good stability properties. The basic idea is to reformulate the original problem eliminating the dependence on the derivative of the solution in the past values. Our hypothesis on the original equation allow us to study the boundedness and asymptotic stability of the true and numerical solutions by the theory of stability with respect to the forcing term.     

General Full Implicit Strong Taylor Approximations for Stiff Stochastic Differential Equations

In this paper, we present the backward stochastic Taylor expansions for a Ito process, including backward Ito-Taylor expansions and backward Stratonovich-Taylor expansions. We construct the general full implicit strong Taylor approximations (including Ito-Taylor and Stratonovich-Taylor schemes) with implicitness in both the deterministic and the stochastic terms for the stiff stochastic differential equations (SSDE) by employing truncations of backward stochastic Taylor expansions. We demonstrate that these schemes will converge strongly with corresponding order $1,2,3,\ldots$ Mean-square stability has been investigated for full implicit strong Stratonovich-Taylor scheme with order $2$, and it has larger mean-square stability region than the explicit and the semi-implicit strong Stratonovich-Taylor schemes with order $2$. We can improve the stability of simulations considerably without too much additional computational effort by using our full implicit schemes. The full implicit strong Taylor schemes allow a larger range of time step sizes than other schemes and are suitable for SSDE with stiffness on both the drift and the diffusion terms. Our numerical experiment shows these points.     

Explicit-implicit difference scheme for the joint solution of the radiative transfer and energy equations by the splitting method

High-order accurate explicit and implicit conservative predictor-corrector schemes are presented for the radiative transfer and energy equations in the multigroup kinetic approximation solved together by applying the splitting method with respect to physical processes and spatial variables. The original system of integrodifferential equations is split into two subsystems: one of partial differential equations without sources and one of ordinary differential equations (ODE) with sources. The general solution of the ODE system and the energy equation is written in quadratures based on total energy conservation in a cell. A feature of the schemes is that a new approximation is used for the numerical fluxes through the cell interfaces. The fluxes are found along characteristics with the interaction between radiation and matter taken into account. For smooth solutions, the schemes approximating the transfer equations on spatially uniform grids are second-order accurate in time and space. As an example, numerical results for Fleck’s test problems are presented that confirm the increased accuracy and efficiency of the method.     

Optimal discrete and continuous mono‐implicit Runge–Kutta schemes for BVODEs

Recent investigations of discretization schemes for the efficient numerical solution of boundary value ordinary differential equations (BVODEs) have focused on a subclass of the well‐known implicit Runge–Kutta (RK) schemes, called mono‐implicit RK (MIRK) schemes, which have been employed in two software packages for the numerical solution of BVODEs, called TWPBVP and MIRKDC. The latter package also employs continuous MIRK (CMIRK) schemes to provide C ¹ continuous approximate solutions. The particular schemes implemented in these codes come, in general, from multi‐parameter families and, in some cases, do not represent optimal choices from these families. In this paper, several optimization criteria are identified and applied in the derivation of optimal MIRK and CMIRK schemes for orders 1–6. In some cases the schemes obtained result from the analysis of existent multi‐parameter families; in other cases new families are derived from which specific optimal schemes are then obtained. New MIRK and CMIRK schemes are presented which are superior to those currently available. Numerical examples are provided to demonstrate the practical improvements that can be obtained by employing the optimal schemes. This revised version was published online in June 2006 with corrections to the Cover Date.     

Exponential stability of numerical solutions for a class of stochastic age-dependent capital system with Poisson jumps

Recently, numerical solutions of stochastic differential equations have received a great deal of attention. Numerical approximation schemes are invaluable tools for exploring their properties. In this paper, we introduce a class of stochastic age-dependent (vintage) capital system with Poisson jumps. We also give the discrete approximate solution with an implicit Euler scheme in time discretization. Using Gronwall’s lemma and Barkholder-Davis-Gundy’s inequality, some criteria are obtained for the exponential stability of numerical solutions to the stochastic age-dependent capital system with Poisson jumps. It is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions, where information on the order of approximation is provided. These error bounds imply strong convergence as the timestep tends to zero. A numerical example is used to illustrate the theoretical results.     

A Half-Explicit Extrapolation Method for Differential-Algebraic Systems of Indix 3

The present paper is concerned with the numerical solution ofdifferential-algebraic index 3 problems. We study an extrapolationmethod based on a variant of Euler's rule. This method onlytreats the algebraic variables implicitly, hence has computationaladvantages compared to fully implicit schemes. A numerical examplein line with the theoretical results is included.