Parameter-uniform Numerical Method Research Articles

Parameter uniform hybrid numerical method for time-dependent singularly perturbed parabolic differential equations with large delay

ABSTRACT In this study, to solve the singularly perturbed delay convection–diffusion–reaction problem, we proposed a hybrid numerical scheme that converges uniformly. Parabolic right boundary layer outcomes from the presence of the small perturbation parameter. To grip this layer behaviour, the problem is solved by Bakhvalov–Shishkin mesh for spatial domain discretization and uniform mesh for temporal domain discretization. A hybrid scheme consisting of a non-polynomial spline scheme for fine mesh and a midpoint upwind scheme for coarse mesh is used to discretize the spatial derivative, while an implicit Euler scheme is used to discretize the time derivative. To make computed solutions more accurate and increase rate of convergence of the scheme, we applied Richardson extrapolation technique. The stability and convergence of the scheme are established. The scheme has a second order of convergence in the discrete supreme norm and is parametric uniformly convergent. The scheme's application is demonstrated through two test problems.

A higher order numerical method for singularly perturbed elliptic problems with characteristic boundary layers

A Petrov-Galerkin finite element method is constructed for a singularly perturbed elliptic problem in two space dimensions. The solution contains a regular boundary layer and two characteristic boundary layers. Exponential splines are used as test functions in one coordinate direction and are combined with bilinear trial functions defined on a Shishkin mesh. The resulting numerical method is shown to be a stable parameter-uniform numerical method that achieves a higher order of convergence compared to upwinding on the same mesh.

A hybrid computational scheme for singularly perturbed Burgers’-Huxley equation

This paper aims to construct and analyze a hybrid computational method for the nonlinear singularly perturbed Burgers’-Huxley equation. The presence of the perturbation parameter and non-linearity in the considered problem makes it difficult to solve the problem analytically and using classical numerical techniques on uniform step sizes as ε goes small. To elucidate such limitations, one can rely on non-classical numerical techniques. In this paper, one such parameter uniform numerical method is designed for the considered problem. The method is constructed:•Firstly, the non-linear terms are linearized via Newton-Raphson-Kantorovich technique. The linearized problem is discretized by the implicit Euler method in the temporal direction.•Secondly, the obtained equation is solved by employing the hybrid computational method comprised of the cubic spline in tension method in the inner layer region and the midpoint upwind method in the outer layer region on a piecewise uniform Shishkin mesh.•Finally, an error analysis of the method is done and observed that the proposed method is parameter uniform convergent with the order of convergence O(τ+N−2ln3N). Three examples are presented and the results are compared to some existing schemes in the literature to demonstrate the reliability of the proposed scheme.

A stable numerical method for singularly perturbed Fredholm integro differential equation using exponentially fitted difference method

This paper implemented a stable numerical scheme for solving singularly perturbed linear second-order Fredholm integro-differential equation. A parameter-uniform numerical method was constructed using an exponentially fitted finite difference method to approximate the differential part and the composite Simpson 1/3 rule for the integral part of the equation. The scheme’s stability and convergence analysis has been carried out. The maximum absolute errors and the rate of convergence are tabulated for different values of the perturbation parameter ɛ and mesh sizes using different numerical test examples.

Fitted finite element methods for singularly perturbed elliptic problems of convection-diffusion type

Fitted finite element methods are constructed for a singularly perturbed convection-diffusion problem in two space dimensions. Exponential splines as basis functions are combined with Shishkin meshes to obtain a stable parameter-uniform numerical method. These schemes satisfy a discrete maximum principle. If the diffusivity parameter is bounded below by some fixed positive constant, the numerical approximations converge, in L∞, at a rate of second order. Moreover, the numerical approximations converge at a rate of first order for all values of the singular perturbation parameter.

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.

Parameter uniform numerical method for a system of singularly perturbed parabolic convection–diffusion equations

This article presents a numerical study of the initial boundary value problem for a singularly perturbed system of two equations of convection–diffusion type. The perturbation parameter in both equations leads to the boundary layer in both solution components. The sign of the convection coefficient decides the position of the boundary layer at the right end of the spatial domain. We suggest a numerical method composed of a spline-based scheme with a Shishkin mesh for solving the proposed system. Convergence analysis shows that the numerical technique is nearly second-order uniformly convergent concerning the perturbation parameter. The numerical illustration is delivered to support the theoretical results.

A parameter-uniform numerical method for singularly perturbed Robin type parabolic convection-diffusion turning point problems

In this work, a parameter-uniform numerical method is constructed for singularly perturbed Robin type parabolic convection-diffusion problems having boundary turning point. The problem is discretized by means of the implicit Euler method in time and the non-standard finite difference method in space on a uniform mesh. Moreover, the non-standard finite difference method is used to discretize the Robin boundary conditions. Via the computation of three numerical examples utilizing the proposed method, the first-order ε-uniform convergence is theoretically demonstrated with regard to space and time directions and numerically proven. The numerical results displayed in tables and graphs supported the theoretical findings. The effectiveness of the proposed method has been verified through some comparisons.

Accelerated parameter-uniform numerical method for singularly perturbed parabolic convection-diffusion problems with a large negative shift and integral boundary condition

The singularly perturbed parabolic convection–diffusion equations with integral boundary conditions and a large negative shift are studied in this paper. The implicit Euler method for the temporal direction and the exponentially fitted finite difference scheme for the spatial direction are applied to formulate a parameter-uniform numerical method. The Simpson’s integration rule is used to handle the integral boundary condition. The Richardson extrapolation technique is applied to enhance the order of convergence of the method. The stability and uniform convergence analysis of the proposed method are studied. It is shown that the method is uniformly convergent with a convergence order of two in both temporal and spatial direction after Richardson extrapolation. Two test examples are considered to verify the validity of the proposed numerical scheme. The obtained numerical results confirm the theoretical estimates. The proposed method provide more accurate results and a higher order of convergence than methods available in the literature.

A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

A novel parameter-uniform numerical method for a singularly perturbed Volterra integro-differential equation

Parameter Uniform Numerical Method for Singularly Perturbed 2D Parabolic PDE with Shift in Space

Singularly perturbed 2D parabolic delay differential equations with the discontinuous source term and convection coefficient are taken into consideration in this paper. For the time derivative, we use the fractional implicit Euler method, followed by the fitted finite difference method with bilinear interpolation for locally one-dimensional problems. The proposed method is shown to be almost first-order convergent in the spatial direction and first-order convergent in the temporal direction. Theoretical results are illustrated with numerical examples.

A parameter-uniform numerical method for singularly perturbed Burgers’ equation

A parameter-uniform numerical method for singularly perturbed Burgers’ equation

Parameter-Uniform Numerical Scheme for Singularly Perturbed Delay Parabolic Reaction Diffusion Equations with Integral Boundary Condition

Numerical computation for the class of singularly perturbed delay parabolic reaction diffusion equations with integral boundary condition has been considered. A parameter-uniform numerical method is constructed via the nonstandard finite difference method for the spatial direction, and the backward Euler method for the resulting system of initial value problems in temporal direction is used. The integral boundary condition is treated using numerical integration techniques. Maximum absolute errors and the rate of convergence for different values of perturbation parameter ε and mesh sizes are tabulated for two model examples. The proposed method is shown to be parameter-uniformly convergent.

Collocation method using artificial viscosity for time dependent singularly perturbed differential–difference equations

A parameter uniform numerical method is presented for solving singularly perturbed time-dependent differential–difference equations with small shifts. To approximate the terms with the shifts, Taylor’s series expansion is used. The resulting singularly perturbed parabolic partial differential equation is solved using an implicit Euler method in temporal direction and cubic B-spline collocation method for the resulting system of ordinary differential equations in spatial direction, and an artificial viscosity is introduced in the scheme using the theory of singular perturbations. The proposed method is shown to be accurate of order OΔt+h2 by preserving ɛ-uniform convergence, where h and Δt denote spatial and temporal step sizes, respectively. Several test examples are solved to demonstrate the effectiveness of the proposed method. The computed numerical results show that the proposed method provides more accurate results than some methods exist in the literature and suitable for solving such problems with little computational effort.

Parameter uniform numerical methods for singularly perturbed delay parabolic differential equations with non-local boundary condition

The motive of this paper is, to develop accurate and parameter uniform numerical method for solving singularly perturbed delay parabolic differential equation with non-local boundary condition exhibiting parabolic boundary layers. Also, the delay term that occurs in the space variable gives rise to interior layer. Fitted operator finite difference method on uniform mesh that uses the procedures of method of line for spatial discretization and backward Euler method for the resulting system of initial value problems in temporal direction is considered. To treat the non-local boundary condition, Simpsons rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, numerical examples are presented and solved for different values of the perturbation parameter. The method is shown to be accurate of O(h2 + △t) . Finally, conclusion of the work is included at the end.

Parameter-uniform numerical method for a two-dimensional singularly perturbed convection–reaction–diffusion problem with interior and boundary layers

We consider a two-dimensional singularly perturbed convection–reaction–diffusion problem that has discontinuities, along lines parallel to x- and y-axes, in the source term, as well as in the convection and reaction coefficients. The coefficient of the highest-order term is a small positive parameter denoted by ε. Due to the discontinuities, the solution exhibits layers in the interior of the domain, in addition to boundary layers. We propose a decomposition of the solution that yields sharp bounds on its derivatives. A finite difference scheme is constructed on an appropriate Shishkin mesh, and it is established that the computed solution is almost first-order, parameter-uniformly convergent. Numerical results are given to support the theoretical results.

Parameter-uniform approximations for a singularly perturbed convection-diffusion problem with a discontinuous initial condition

A singularly perturbed parabolic problem of convection-diffusion type with a discontinuous initial condition is examined. A particular complimentary error function is identified which matches the discontinuity in the initial condition. The difference between this analytical function and the solution of the parabolic problem is approximated numerically. A coordinate transformation is used so that a layer-adapted mesh can be aligned to the interior layer present in the solution. Numerical analysis is presented for the associated numerical method, which establishes that the numerical method is a parameter-uniform numerical method. Numerical results are presented to illustrate the pointwise error bounds established in the paper.

Extended cubic B-spline collocation method for singularly perturbed parabolic differential-difference equation arising in computational neuroscience.

A parameter uniform numerical method is presented for solving singularly perturbed parabolic differential-difference equations with small shift arguments in the reaction terms arising in computational neuroscience. To approximate the terms with the shift arguments, Taylor's series expansion is used. The resulting singularly perturbed parabolic differential equation is solved by applying the implicit Euler method in temporal direction and extended cubic B-spline basis functions consisting of a free parameter λ for the resulting system of ordinary differential equations in the spatial direction. The proposed method is shown to be accurate of order by preserving an ε- uniform convergence. To demonstrate the applicability of the proposed method, two test examples are solved by the method and the numerical results are compared with some existing results. The obtained numerical results agreed with the theoretical results.

Parameter-uniform approximation on equidistributed meshes for singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions

In this work we develop a parameter-uniform numerical method on equidistributed meshes for solving a class of singularly perturbed parabolic reaction-diffusion problems with Robin boundary conditions. The discretization consists of a modified Euler scheme in time, a central difference scheme in space, and a special finite difference scheme for the Robin boundary conditions. A uniform mesh is used in the time direction while the mesh in the space direction is generated via the equidistribution of a suitably chosen monitor function. We discuss error analysis and prove that the method is parameter-uniformly convergent of order two in space and order one in time. To support the theoretical result, some numerical experiments are performed.

A higher order numerical scheme for singularly perturbed parabolic turning point problems exhibiting twin boundary layers

In this article, a parameter-uniform numerical method is presented to solve one-dimensional singularly perturbed parabolic convection-diffusion multiple turning point problems exhibiting two exponential boundary layers. We study the asymptotic behaviour of the solution and its partial derivatives. The problem is discretized using the implicit Euler method for time discretization on a uniform mesh and a hybrid scheme for spatial discretization on a generalized Shishkin mesh. The scheme is shown to be ε-uniformly convergent of order one in time direction and order two in spatial direction upto a logarithmic factor. Numerical experiments are conducted to validate the theoretical results. Comparison is done with the upwind scheme on a uniform mesh as well as on the standard Shishkin mesh to demonstrate the higher order accuracy of the proposed scheme on a generalized Shishkin mesh.