Convergence Of Numerical Scheme Research Articles

Exponentially convergent numerical scheme for the stream function of potential flow over axisymmetric bodies

A boundary element scheme for the problem of potential flow over an axisymmetric toroidal body is considered. An integral equation for the velocity distribution on the body is derived. It is shown that the numerical scheme for solving the considered equation converges exponentially.

Exponentially Convergent Numerical Scheme for the Stream Function of Potential Flow over Axisymmetric Bodies

Exponentially Convergent Numerical Scheme for the Stream Function of Potential Flow over Axisymmetric Bodies

A positivity-preserving, linear, energy stable and convergent numerical scheme for the Poisson–Nernst–Planck (PNP) system

This article focuses on the convergence analysis for a fully discrete finite difference scheme for the time-dependent Poisson–Nernst–Planck system. The numerical scheme, a three-level linearized finite difference algorithm based on the reformulation of the Nernst–Planck equations, which was developed in [J. Sci. Comput. 81(2019),436–458]. The positivity-preserving property and energy stability were theoretically established. In this paper, we rigorously prove first-order convergence in time and second-order convergence in space for the numerical scheme in the ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1) norm under the condition that time step size is linearly proportional to spatial mesh size, in which the natural property of the exponential terms gives a lot of challenges. Moreover, the higher-order asymptotic expansion (up to third-order temporal accuracy and fourth-order spatial accuracy) has to be involved, due to the leading local truncation error will not be enough to recover an ℓ∞ bound for ion concentrations n and p. To our knowledge, this scheme will be the first linear decoupled algorithm to combine the following theoretical properties for the PNP system: ion concentration positivity preserving, unconditionally energy stability and optimal rate convergence. Numerical results are shown to be consistent with theoretical analysis.

A fitted operator numerical method for singularly perturbed Fredholm integro-differential equation with integral initial condition

ObjectivesIn this paper, a uniformly convergent numerical scheme is proposed for solving a singularly perturbed Fredholm integro-differential equation with an integral initial condition. The equation involves a left boundary layer which makes it difficult to solve it using the standard numerical methods. A fitted operator finite difference method is used to approximate the differential part of the equation and the composite Simpson 13\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\frac{1}{3}$$\\end{document} rule is used for the integral parts of the equation and the initial condition.ResultThe stability bound and error estimation of the approximated solution are performed, to show the uniform convergence of the scheme with order one in the maximum norm. Numerical test examples are provided to calculate the maximum absolute errors, thrgence, and the uniform error for a couple of examples to support theoretical analysis.

Fitted Numerical Scheme for Singularly Perturbed Convection‐Diffusion Equation with Small Time Delay

In this article, a uniformly convergent numerical scheme is developed to solve a singularly perturbed convection‐diffusion equation with a small delay having a boundary layer along the left side. A priori bounds of continuous solution and its derivatives are discussed. To solve the problem, the Crank–Nicolson scheme in the time direction and the exponentially fitted finite difference scheme in the space direction are used. The stability of the method is analyzed. It is proved that the developed scheme converges uniformly with first order in space and second order in time. To validate the applicability of the theoretical finding of the developed scheme, numerical experiments are carried out by considering two test examples.

On the optimally controlled stochastic shallow lake

We consider the stochastic control problem of the shallow lake and continue the work of Kossioris, Loulakis, and Souganidis. First, we generalise the characterisation of the value function as the viscosity solution of a well-posed problem to include more general recycling rates. Then, we prove approximate optimality under bounded controls and we establish quantitative estimates. Finally, we implement a convergent and stable numerical scheme for the computation of the value function to investigate properties of the optimally controlled stochastic shallow lake. This approach permits to derive tails asymptotics for the invariant distribution and to extend results of Grass, Kiseleva and Wagener. https://www.sciencedirect.com/science/article/pii/S1007570414004754] beyond the small noise limit.

Uniformly convergent numerical method for time-fractional convection–diffusion equation with variable coefficients

This paper presents a uniformly convergent numerical scheme for singularly perturbed fractional order convection–diffusion equations with variable coefficients. First, the time-fractional derivative is considered in the Caputo sense and treated using the implicit Euler method. Then, a uniformly convergent numerical scheme based on the nonstandard finite difference method is developed. The technique is proved rigorously for parameter-uniform convergence. By numerical experimentation, it is also validated that the computational results agree with theoretical results.

Uniformly convergent extended cubic B-spline collocation method for two parameters singularly perturbed time-delayed convection-diffusion problems

This work proposes a uniformly convergent numerical scheme to solve singularly perturbed parabolic problems of large time delay with two small parameters. The approach uses implicit Euler and the exponentially fitted extended cubic B-spline for time and space derivatives respectively. Extended cubic B-splines have advantages over classical B-splines. This is because for a given value of the free parameter lambda the solution obtained by the extended B-spline is better than the solution obtained by the classical B-spline. To confirm the correspondence of the numerical methods with the theoretical results, numerical examples are presented. The present numerical technique converges uniformly, leading to the current study of being more efficient.

Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation

This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results.

Fitted computational method for singularly perturbed convection-diffusion equation with time delay

A uniformly convergent numerical scheme is proposed to solve a singularly perturbed convection-diffusion problem with a large time delay. The diffusion term of the problem is multiplied by a perturbation parameter, ε. For a small ε, the problem exhibits a boundary layer, which makes it challenging to solve it analytically or using standard numerical methods. As a result, the backward Euler scheme is applied in the temporal direction. Non-symmetric finite difference schemes are applied for approximating the first-order derivative terms, and a higher-order finite difference method is applied for approximating the second-order derivative term. Furthermore, an exponential fitting factor is computed and induced in the difference scheme to handle the effect of the small parameter. Using the discrete maximum principle, the stability of the scheme is examined and analyzed. The developed scheme is parameter-uniform with a linear order of convergence in both space and time. To examine the accuracy of the method, two model examples are considered. Further, the boundary layer behavior of the solutions is given graphically.

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

Uniformly Convergent Numerical Scheme for Solving Singularly Perturbed Parabolic Convection-Diffusion Equations with Integral Boundary Condition

A tension spline fitted numerical scheme for singularly perturbed reaction-diffusion problem with negative shift

ObjectiveThe paper is focused on developing and analyzing a uniformly convergent numerical scheme for a singularly perturbed reaction-diffusion problem with a negative shift. The solution of such problem exhibits strong boundary layers at the two ends of the domain due to the influence of the perturbation parameter, and the term with negative shift causes interior layer. The rapidly changing behavior of the solution in the layers brings significant difficulties in solving the problem analytically. We have treated the problem by proposing a numerical scheme using the implicit Euler method in the temporal direction and a fitted tension spline method in the spatial direction with uniform meshes.ResultStability and uniform error estimates are investigated for the developed numerical scheme. The theoretical finding is demonstrated by numerical examples. It is obtained that the developed numerical scheme is uniformly convergent of order one in time and order two in space.

High order accurate and convergent numerical scheme for the strongly anisotropic Cahn–Hilliard model

AbstractWe propose and analyze a second order accurate in time, energy stable numerical scheme for the strongly anisotropic Cahn–Hilliard system, in which a biharmonic regularization has to be introduced to make the equation well‐posed. A convexity analysis on the anisotropic interfacial energy is necessary to overcome an essential difficulty associated with its highly nonlinear and singular nature. The second order backward differentiation formula temporal approximation is applied, combined with Fourier pseudo‐spectral spatial discretization. The nonlinear surface energy part is updated by an explicit extrapolation formula. Meanwhile, the energy stability analysis is enforced by the fact that all the second order functional derivatives of the energy stay uniformly bounded by a global constant. A Douglas‐Dupont type regularization is added to stabilize the numerical scheme, and a careful estimate ensures a modified energy stability with a uniform constraint for the regularization parameter . In turn, the combination with an appropriate treatment for the nonlinear double well potential terms leads to a weakly nonlinear scheme. More importantly, such an energy stability is in terms of the interfacial energy with respect to the original phase variable, which enables us to derive an optimal rate convergence analysis.

Uniformly convergent hybrid numerical scheme for singularly perturbed turning point problems with delay

In the present work, we consider a class of singularly perturbed turning point problems with delay for their numerical solution. Due to the presence of a delay, there occurs an interior layer in the solution along with twin boundary layers (SIAM J. Appl. Math. 42(3): 502–530). Some a priori estimates are given on the exact solution and its derivatives. A numerical technique composed of hybrid finite difference scheme on a layer-adapted Shishkin mesh is proposed. We establish the consistency, stability and convergence of the proposed technique. To validate the theoretical predictions, we present some numerical illustrations. The proposed scheme is proved to have an almost second-order convergence rate, uniform with respect to the perturbation parameter ε.

Error analysis for Galerkin-BDF discretizations of DAEs with elliptic operator constraints

We are interested in a convergent numerical discretization scheme for nonlinear differential algebraic equations (DAEs) coupled with elliptic constraints. The dynamics of flow networks (for example circuits) can often be described by a DAE. However, this is only the case if all network element models are lumped models. If spatial effects cannot be neglected, distributed element models have to be included. In this paper, we address the case of elliptic models for distributed network elements. Under some monotonicity and Lipschitz assumptions for the system functions, we present an error analysis for discretizations of DAEs coupled with elliptic constraints based on a Galerkin approach for the distributed model part combined with the BDF method as time discretization of the full system.

A uniformly convergent numerical scheme for solving singularly perturbed differential equations with large spatial delay

In this study, a parameter-uniform numerical scheme is built and analyzed to treat a singularly perturbed parabolic differential equation involving large spatial delay. The solution of the considered problem has two strong boundary layers due to the effect of the perturbation parameter, and the large delay causes a strong interior layer. The behavior of the layers makes it difficult to solve such problem analytically. To treat the problem, we developed a numerical scheme using the weighted average (theta-method) difference approximation on a uniform time mesh and the central difference method on a piece-wise uniform spatial mesh. We established the Stability and convergence analysis for the proposed scheme and obtained that the method is uniformly convergent of order two in the temporal direction and almost second order in the spatial direction. To validate the applicability of the proposed numerical scheme, two model examples are treated and confirmed with the theoretical findings.

Numerical scheme for Erdélyi–Kober fractional diffusion equation using Galerkin–Hermite method

The aim of this work is to devise and analyse an accurate numerical scheme to solve Erdélyi–Kober fractional diffusion equation. This solution can be thought as the marginal pdf of the stochastic process called the generalized grey Brownian motion (ggBm). The ggBm includes some well-known stochastic processes: Brownian motion, fractional Brownian motion, and grey Brownian motion. To obtain a convergent numerical scheme we transform the fractional diffusion equation into its weak form and apply the discretization of the Erdélyi–Kober fractional derivative. We prove the stability of the solution of the semi-discrete problem and its convergence to the exact solution. Due to the singularity in the time term appearing in the main equation, the proposed method converges slower than first order. Finally, we provide a numerical analysis of the full-discrete problem using orthogonal expansion in terms of Hermite functions.

An Exponentially Fitted Numerical Scheme via Domain Decomposition for Solving Singularly Perturbed Differential Equations with Large Negative Shift

In this study, we focus on the formulation and analysis of an exponentially fitted numerical scheme by decomposing the domain into subdomains to solve singularly perturbed differential equations with large negative shift. The solution of problem exhibits twin boundary layers due to the presence of the perturbation parameter and strong interior layer due to the large negative shift. The original domain is divided into six subdomains, such as two boundary layer regions, two interior (interfacing) layer regions, and two regular regions. Constructing an exponentially fitted numerical scheme on each boundary and interior layer subdomains and combining with the solutions on the regular subdomains, we obtain a second order ε -uniformly convergent numerical scheme. To demonstrate the theoretical results, numerical examples are provided and analyzed.

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α > 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 < α < 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

A Finite Difference Method for the Variational p-Laplacian

We propose a new monotone finite difference discretization for the variational p-Laplace operator, Delta _pu=text{ div }(|nabla u|^{p-2}nabla u), and present a convergent numerical scheme for related Dirichlet problems. The resulting nonlinear system is solved using two different methods: one based on Newton-Raphson and one explicit method. Finally, we exhibit some numerical simulations supporting our theoretical results. To the best of our knowledge, this is the first monotone finite difference discretization of the variational p-Laplacian and also the first time that nonhomogeneous problems for this operator can be treated numerically with a finite difference scheme.