Monotone Finite Difference Scheme Research Articles

An efficient numerical scheme for 1D parabolic singularly perturbed problems with an interior and boundary layers

In this paper we consider a 1D parabolic singularly perturbed reaction-convection–diffusion problem, which has a small parameter in both the diffusion term (multiplied by the parameter ε2) and the convection term (multiplied by the parameter μ) in the differential equation (ε∈(0,1], μ∈[0,1], μ≤ε). Moreover, the convective term degenerates inside the spatial domain, and also the source term has a discontinuity of first kind on the degeneration line. In general, for sufficiently small values of the diffusion and the convection parameters, the exact solution exhibits an interior layer in a neighborhood of the interior degeneration point and also a boundary layer in a neighborhood of both end points of the spatial domain. We study the asymptotic behavior of the exact solution with respect to both parameters and we construct a monotone finite difference scheme, which combines the implicit Euler method, defined on a uniform mesh, to discretize in time, together with the classical upwind finite difference scheme, defined on an appropriate nonuniform mesh of Shishkin type, to discretize in space. The numerical scheme converges in the maximum norm uniformly in ε and μ, having first order in time and almost first order in space. Illustrative numerical results corroborating in practice the theoretical results are showed.

SCHEMES CONVERGENT Ε-UNIFORMLY FOR PARABOLIC SINGULARLY PERTURBED PROBLEMS WITH A DEGENERATING CONVECTIVE TERM AND A DISCONTINUOUS SOURCE

We consider the numerical approximation of a 1D singularly perturbed convection-diffusion problem with a multiply degenerating convective term, for which the order of degeneracy is 2p + 1, p is an integer with p ≥ 1, and such that the convective flux is directed into the domain. The solution exhibits an interior layer at the degeneration point if the source term is also a discontinuous function at this point. We give appropriate bounds for the derivatives of the exact solution of the continuous problem, showing its asymptotic behavior with respect to the perturbation parameter ε, which is the diffusion coefficient. We construct a monotone finite difference scheme combining the implicit Euler method, on a uniform mesh, to discretize in time, and the upwind finite difference scheme, constructed on a piecewise uniform Shishkin mesh condensing in a neighborhood of the interior layer region, to discretize in space. We prove that the method is convergent uniformly with respect to the parameter ε, i.e., ε-uniformly convergent, having first order convergence in time and almost first order in space. Some numerical results corroborating the theoretical results are showed.

Efficient parameter estimation in a macroscopic traffic flow model by discrete mollification

Our concern is the numerical identification of traffic flow parameters in a macroscopic one-dimensional model whose governing equation is strongly degenerate parabolic. The unknown parameters determine the flux and the diffusion terms. The parameters are estimated by repeatedly solving the corresponding direct problem under variation of the parameter values, starting from an initial guess, with the aim of minimizing the distance between a time-dependent observation and the corresponding numerical solution. The direct problem is solved by a modification of a well-known monotone finite difference scheme obtained by discretizing the nonlinear diffusive term by a formula that involves a discrete mollification operator. The mollified scheme occupies a larger stencil but converges under a less restrictive Courant-Friedrichs-Lewy (CFL) condition, which allows one to employ a larger time step. The ability of the proposed procedure for the identification of traffic flow parameters is illustrated by a numerical experiment.

Another uniform convergence analysis technique of some numerical methods for parabolic singularly perturbed problems

In this work we analyze the uniform convergence of some monotone finite difference schemes which are used to semidiscretize in space time dependent two dimensional singularly perturbed parabolic problems of convection–diffusion or reaction–diffusion type. The analysis technique combines a suitable semidiscrete maximum principle joint to some techniques used in the study of the convergence of well known numerical schemes developed to solve elliptic singularly perturbed problems. We focus our attention to standard central or upwind finite difference schemes defined on special nonuniform meshes of Shishkin type. Nevertheless, the technique can be easily extended to other classical discretization methods like, for example, fitted operator methods. We prove that the stiff initial value problems resulting of the spatial semidiscretization processes, have a unique solution which converges uniformly with respect to the singular perturbation parameter. To obtain an efficient numerical algorithm, such initial value problems are discretized by using appropriate time integrators; here, we have chosen the Implicit Euler method for doing this, obtaining an unconditional and uniformly convergent numerical method in a simple way. Finally, some numerical results are shown in order to illustrate the efficiency of the numerical methods, according to the theoretical results.

𝐿¹ error estimates for difference approximations of degenerate convection-diffusion equations

We analyze monotone finite difference schemes for strongly degenerate convection-diffusion equations in one spatial dimension. These nonlinear equations are well-posed within a class of (discontinuous) entropy solutions. We prove that the L 1 L^1 error between the approximate and exact solutions is O ( Δ x 1 / 3 ) \mathcal {O}(\Delta x^{1/3}) , where Δ x \Delta x is the spatial grid parameter. This result should be compared with the classical O ( Δ x 1 / 2 ) \mathcal {O}(\Delta x^{1/2}) error estimate for conservation laws (Kuznecov, 1976), and a recent estimate of O ( Δ x 1 / 11 ) \mathcal {O}(\Delta x^{1/11}) for degenerate convection-diffusion equations (Karlsen, Koley, Risebro 2012).

Conditioning and stability of finite difference schemes on uniform meshes for a singularly perturbed parabolic convection-diffusion equation

For a singularly perturbed parabolic convection-diffusion equation, the conditioning and stability of finite difference schemes on uniform meshes are analyzed. It is shown that a convergent standard monotone finite difference scheme on a uniform mesh is not ɛ-uniformly well conditioned or ɛ-uniformly stable to perturbations of the data of the grid problem (here, ɛ is a perturbation parameter, ɛ ∈ (0, 1]). An alternative finite difference scheme is proposed, namely, a scheme in which the discrete solution is decomposed into regular and singular components that solve grid subproblems considered on uniform meshes. It is shown that this solution decomposition scheme converges ɛ-uniformly in the maximum norm at an O(N−1lnN + N0−1) rate, where N + 1 and N0 + 1 are the numbers of grid nodes in x and t, respectively. This scheme is ɛ-uniformly well conditioned and ɛ-uniformly stable to perturbations of the data of the grid problem. The condition number of the solution decomposition scheme is of order O(δ−2lnδ−1 + δ0−1); i.e., up to a logarithmic factor, it is the same as that of a classical scheme on uniform meshes in the case of a regular problem. Here, δ = N−1lnN and δ0 = N0−1 are the accuracies of the discrete solution in x and t, respectively.

Monotone difference schemes stabilized by discrete mollification for strongly degenerate parabolic equations

AbstractThe discrete mollification method is a convolution‐based filtering procedure suitable for the regularization of ill‐posed problems and for the stabilization of explicit schemes for the solution of PDEs. This method is applied to the discretization of the diffusive terms of a known first‐order monotone finite difference scheme [Evje and Karlsen, SIAM J Numer Anal 37 (2000) 1838–1860] for initial value problems of strongly degenerate parabolic equations in one space dimension. It is proved that the mollified scheme is monotone and converges to the unique entropy solution of the initial value problem, under a CFL stability condition which permits to use time steps that are larger than with the unmollified (basic) scheme. Several numerical experiments illustrate the performance and gains in CPU time for the mollified scheme. Applications to initial‐boundary value problems are included. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 38–62, 2012

Numerical analysis and computing of a non-arbitrage liquidity model with observable parameters for derivatives

This paper deals with the numerical analysis and computing of a nonlinear model of option pricing appearing in illiquid markets with observable parameters for derivatives. A consistent monotone finite difference scheme is proposed and a stability condition on the stepsize discretizations is given.

A singular perturbation approach to solve Burgers–Huxley equation via monotone finite difference scheme on layer-adaptive mesh

This paper deals with a numerical method for solving one-dimensional unsteady Burgers–Huxley equation with the viscosity coefficient ε. The parameter ε takes any values from the half open interval (0, 1]. At small values of the parameter ε, an outflow boundary layer is produced in the neighborhood of right part of the lateral surface of the domain and the problem can be considered as a non-linear singularly perturbed problem with a singular perturbation parameter ε. Using singular perturbation analysis, asymptotic bounds for the derivatives of the solution are established by decomposing the solution into smooth and singular components. We construct a numerical scheme that comprises of implicit-Euler method to discretize in temporal direction on uniform mesh and a monotone hybrid finite difference operator to discretize the spatial variable with piecewise uniform Shishkin mesh. To obtain better accuracy, we use central finite difference scheme in the boundary layer region. Shishkin meshes are refined in the boundary layer region, therefore stability constraint is satisfied by proposed scheme. Quasilinearization process is used to tackle the non-linearity and it is shown that quasilinearization process converges quadratically. The method has been shown to be first order uniformly accurate in the temporal variable, and in the spatial direction it is first order parameter uniform convergent in the outside region of boundary layer, and almost second order parameter uniform convergent in the boundary layer region. Accuracy and uniform convergence of the proposed method is demonstrated by numerical examples and comparison of numerical results made with the other existing methods.

Numerical analysis and computing for option pricing models in illiquid markets

Nowadays market liquidity has become an issue of very high concern in financial risk management. This paper deals with the numerical analysis and computing of nonlinear models of option pricing that appear when illiquid market effects are taken into account. A consistent monotone finite difference scheme is proposed and a relationship between the discretization step size is obtained, ensuring nonnegative and stable numerical solutions and avoiding spurious oscillations.

Optimal Control of the Propagation of a Graph in Inhomogeneous Media

We study an optimal control problem for viscosity solutions of a Hamilton-Jacobi equation describing the propagation of a one-dimensional graph with the control being the speed function. The existence of an optimal control is proved together with an approximate controllability result in the $H^{-1}$-norm. We prove convergence of a discrete optimal control problem based on a monotone finite difference scheme and describe some numerical results.

Wide stencil finite difference schemes for the elliptic Monge-Ampère equation and functions of the eigenvalues of the Hessian

Certain fully nonlinear elliptic Partial Differential Equations can be written as functions of the eigenvalues of the Hessian. These include: the Monge-Ampere equation, Pucci’s Maximal and Minimal equations, and the equation for the convex envelope. In this article we build convergent monotone finite difference schemes for the aforementioned equations. Numerical results are presented.

On The Parallel Multiblock Geometric Multigrid Algorithm

Abstract The application of a parallel multiblock geometric multigrid is consid-ered. It is applied to solve a two-dimensional poroelastic model. This system of PDEs is approximated by a special stabilized monotone finite-difference scheme. The obtained system of linear algebraic equations is solved by a multigrid method, when a domain is partitioned into structured blocks. A new strategy for the solution of the discrete problem on the coarsest grid is proposed and the efficiency of the obtained algorithm is investigated. The geometrical structure of the sequential multigrid method is used to develop a parallel version of the multigrid algorithm. The convergence properties of several smoothers are investigated and some computational results are presented.

On an upwind difference scheme for strongly degenerate parabolic equations modelling the settling of suspensions in centrifuges and non-cylindrical vessels

We prove the convergence of an explicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two zero-flux boundary conditions. This problem arises in a model of sedimentation–consolidation processes in centrifuges and vessels with varying cross-sectional area. We formulate the definition of entropy solution of the model in the sense of Kružkov and prove the convergence of the scheme to the unique BV entropy solution of the problem. The scheme and the model are illustrated by numerical examples.

Approximations for Viscosity Solutions of Hamilton--Jacobi Equations With Locally Varying Time and Space Grids

A new monotone finite difference scheme is introduced that approximates viscosity solutions of first-order nonlinear Hamilton--Jacobi equations. The main feature of the scheme is that it allows for locally varying time and space grids, making it ideal for use with adaptive algorithms. Explicit a priori error estimates are given to establish convergence. Numerical examples, including adaptive mesh refinement examples, demonstrate the effectiveness of the proposed scheme.

A semi-implicit monotone difference scheme for an initial-boundary value problem of a strongly degenerate parabolic equation modeling sedimentation-consolidation processes

We prove the convergence of a semi-implicit monotone finite difference scheme approximating an initial-boundary value problem for a spatially one-dimensional quasilinear strongly degenerate parabolic equation, which is supplied with two different inhomogeneous flux-type boundary conditions. This problem arises in the modeling of the sedimentation-consolidation process. We formulate the definition of entropy solution of the model in the sense of Kru z ˇ \check {\mbox {z}} kov and prove convergence of the scheme to the unique B V BV entropy solution of the problem, up to satisfaction of one of the boundary conditions.

Grid approximation of a singularly perturbed boundary value problem modelling heat transfer in the case of flow over a flat plate with suction of the boundary layer

In the present paper we consider a boundary value problem on the semiaxis (0,∞) for a singularly perturbed parabolic equation with the two perturbation parameters ε 1 and ε 2 multiplying, respectively, the second and first derivatives with respect to the space variable. Depending on the relation between the parameters, the differential equation can be either of reaction–diffusion type or of convection–diffusion type. Correspondingly, the boundary layer can be either parabolic or regular. For this problem we consider the case when the boundary layer can be controlled by continuous suction of the fluid out of the boundary layer (model problems of this type appear in the mathematical modelling of heat transfer processes for flow past a flat plate). Errors in the approximations generated by standard numerical methods can be unsatisfactorily large for small values of the parameter ε 1. We construct a monotone finite difference scheme on piecewise uniform meshes which generates numerical solutions converging ε-uniformly with order O(N −1 ln N+N 0 −1) , where N 0 is the number of nodes in the time mesh and N is the number of meshpoints on a unit interval of the semiaxis in x. Although the solution of problem has a singularity only for ε 1→0, the character of the boundary layer depends essentially on the vector-valued parameter ε=( ε 1, ε 2). This prevents us from constructing an ε-uniformly convergent scheme having a transition parameter which is independent of the parameter ε 2.

ON A DIFFUSIVELY CORRECTED KINEMATIC-WAVE TRAFFIC FLOW MODEL WITH CHANGING ROAD SURFACE CONDITIONS

The well-known Lighthill–Whitham–Richards kinematic traffic flow model for unidirectional flow on a single-lane highway is extended to include both abruptly changing road surface conditions and drivers' reaction time and anticipation length. The result is a strongly degenerate convection–diffusion equation, where the diffusion term, accounting for the drivers' behavior, is effective only where the local car density exceeds a critical value, and the convective flux function depends discontinuously on the location. It is shown that the validity of the proposed traffic model is supported by a recent mathematical well-posedness (existence and uniqueness) theory for quasilinear degenerate parabolic convection–diffusion equations with discontinuous coefficients.20,22This theory includes a convergence proof for a monotone finite-difference scheme, which is used herein to simulate the traffic flow model for a variety of situations.

Monotone Finite Difference Schemes for Nonlinear Systems with Mixed Quasi-Monotonicity

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.

The Optimal Convergence Rate of Monotone Finite Difference Methods for Hyperbolic Conservation Laws

We are interested in the rate of convergence in L1 of the approximate solution of a conservation law generated by a monotone finite difference scheme. Kuznetsov has proved that this rate is 1/2 [USSR Comput. Math. Math. Phys., 16 (1976), pp. 105--119 and Topics Numer. Anal. III, in Proc. Roy. Irish Acad. Conf., Dublin, 1976, pp. 183--197], and recently Teng and Zhang have proved this estimate to be sharp for a linear flux [SIAM J. Numer. Anal., 34 (1997), pp. 959--978]. We prove, by constructing appropriate initial data for the Cauchy problem, that Kuznetsov's estimates are sharp for a nonlinear flux as well.