Richardson Extrapolation Technique Research Articles

A sixth-order finite difference method for the two-dimensional nonlinear advection diffusion reaction equation

A novel higher-order finite difference method is developed to solve the two-dimensional nonlinear advection diffusion reaction equation. Firstly, a new five-point sixth-order finite difference scheme is proposed to discretize spatial derivatives and the sixth-order backward difference method is applied to teat temporal derivative. As a result, a new fully implicit scheme with sixth-order accuracy in both time and space is proposed. Subsequently, for the calculation of the starting step of the sixth-order backward difference method, the Crank-Nicolson method combined with the Richardson extrapolation technique is employed to obtain the equivalent temporal accuracy as the main scheme. Finally, the accuracy and reliability of the presented method are validated through various numerical examples.

An efficient numerical approach for singularly perturbed time delayed parabolic problems with two-parameters

ObjectivesThe main objective of this work is to design an efficient numerical scheme is proposed for solving singularly perturbed time delayed parabolic problems with two parameters.ResultsThe scheme is constructed via the implicit Euler and non-standard finite difference method to approximate the time and space derivatives, respectively. Besides, to enhance the accuracy and order of convergence of the method Richardson extrapolation technique is employed. Intensive numerical experimentation has been done on some model examples. Further, the layer behavior of the solutions is presented using graphs and observed to agree with the existing theories. Finally, error analysis of the scheme is done and observed that the proposed method is parameter uniform convergent with the order of convergence (Δt)2+h2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\left( (\\Delta t )^2+h^2\\right) $$\\end{document}

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

A Novel Richardson Extrapolation Technique on a Shishkin and Exponentially Graded Mesh for Singularly Perturbed Convection-diffusion Problems With Integral Boundary Conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson extrapolation technique for singularly perturbed convection-diffusion problem with non-local boundary conditions

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

Richardson Extrapolation Technique for Singularly Perturbed Parabolic Convection-Diffusion Problems With A Discontinuous Initial Condition

Fitted mesh scheme for singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers

In this paper, fitted mesh numerical scheme is presented for solving singularly perturbed parabolic convection–diffusion problem exhibiting twin boundary layers. To approximate the solution, we discretize the temporal variable on uniform mesh and discretize the spatial one on piecewise uniform mesh of the Shishkin mesh type. The resulting scheme is shown to be almost first order convergent that accelerated to almost second order convergent by applying the Richardson extrapolation technique. Stability and consistency of the proposed method are established very well in order to guarantee the convergence of the method. Further, the theoretical investigations are confirmed by numerical experiments. Moreover, the present scheme is stable, consistent and gives more accurate solution than existing methods in the literature.

Richardson Extrapolation Technique for Singularly Perturbed Convection-diffusion Problem with Non-local Boundary Conditions

This study presents the Richardson extrapolation techniques for solving singularly perturbed convection -diffusion problems (SPCDP) with non-local boundary conditions. A numerical approach is presented using an upwind finite difference scheme a piecewise-uniform (Shishkin) mesh. To handle the non-local boundary conditions, the trapezoidal rule is applied. The study establishes an error bound for numerical solutions and determines the numerical approximation for scaled derivatives. To enhance convergence and accuracy, we utilize Richardson extrapolation. This elevates accuracy from firs-order to second order convergence

BE-BDF2 Time Integration Scheme Equipped with Richardson Extrapolation for Unsteady Compressible Flows

In this work we investigate the effectiveness of the Backward Euler-Backward Differentiation Formula (BE-BDF2) in solving unsteady compressible inviscid and viscous flows. Furthermore, to improve its accuracy and its order of convergence, we have equipped this time integration method with the Richardson Extrapolation (RE) technique. The BE-BDF2 scheme is a second-order accurate, A-stable, L-stable and self-starting scheme. It has two stages: the first one is the simple Backward Euler (BE) and the second one is a second-order Backward Differentiation Formula (BDF2) that uses an intermediate and a past solution. The RE is a very simple and powerful technique that can be used to increase the order of accuracy of any approximation process by eliminating the lowest order error term(s) from its asymptotic error expansion. The spatial approximation of the governing Navier–Stokes equations is performed with a high-order accurate discontinuous Galerkin (dG) method. The presented numerical results for canonical test cases, i.e., the isentropic convecting vortex and the unsteady vortex shedding behind a circular cylinder, aim to assess the performance of the BE-BDF2 scheme, in its standard version and equipped with RE, by comparing it with the ones obtained by using more classical methods, like the BDF2, the second-order accurate Crank–Nicolson (CN2) and the explicit third-order accurate Strong Stability Preserving Runge–Kutta scheme (SSP-RK3).

Shear locking free polygonal elements for the analysis of functionally graded plates using (n + 1) integration scheme and Reissner-Mindlin theory

A new shear-locking free polygonal elements is proposed for static bending and free vibration analysis of thin/thick functionally graded material plates. The domain is discretized with arbitrary polygons and Wachspress interpolants are employed for approximating the unknowns. The plate kinematics is based on Reissner-Mindlin plate theory and shear locking syndrome is suppressed by employing the novel n + 1 integration scheme based on Richardson extrapolation technique. Within this, a two stage sampling technique is introduced with the Weighted Richardson scheme to compute the system matrices. From the systematic study, it is inferred that the proposed scheme passes patch tests and yields accurate results.

An improved time accurate numerical estimation for singularly perturbed semilinear parabolic differential equations with small space shifts and a large time lag

The objective of this work is to provide a robust numerical scheme for solving a singularly perturbed semilinear parabolic differential–difference equation having both time delay along with spatial delay, and shift parameters. The small delay and shift arguments in spatial direction are treated with Taylor’s approximation. The technique of quasilinearization is used to treat the semilinearity and the temporal direction is handled with the θ-method. To handle the layer behavior caused by the presence of perturbation parameter, the problem is solved using the upwind scheme on two-layer resolving meshes in the spatial direction providing a first-order accurate result for 0.5≤θ≤1. For θ=0.5, we have the second-order accurate Crank–Nicolson scheme in time. In order to have a higher-order accuracy, the Richardson extrapolation technique is applied in the spatial direction, which in turn provides a global second-order accurate result. Thomas algorithm is used to solve the tridiagonal system of equations formed in the fully discrete scheme. The numerical approach presented is shown to be parameter uniform and convergent for 0.5≤θ≤1. Some numerical tests are presented to show the efficacy of the proposed scheme.

Richardson extrapolation technique for generalized Black–Scholes PDEs for European options

Richardson extrapolation technique for generalized Black–Scholes PDEs for European options

Parameter robust higher‐order finite difference method for convection‐diffusion problem with time delay

AbstractThis paper deals with the study of a higher‐order numerical approximation for a class of singularly perturbed convection‐diffusion problems with time delay. The method combines a higher‐Order Difference with an Identity Expansion (HODIE) scheme over a piece‐wise uniform mesh in the spatial direction and the backward Euler method on a uniform mesh for discretization in the temporal direction. A priori bounds for the continuous solution and its derivatives are derived by splitting the solution into regular and singular components. These bounds are useful in the error analysis of the proposed scheme. The present scheme converges ‐uniformly with the order of convergence one in time and almost second‐order in space direction. Further, to increase the rate of convergence in the time variable, we implemented the Richardson extrapolation technique. Thus, finally, the resultant scheme with Richardson extrapolation in time becomes almost second‐order ‐uniformly convergent in both the space and time variable. The detailed stability and convergence analysis have been done using the derived a priori estimates. We consider three test problems to validate the predicted theory and show that numerical results are in good agreement with our theoretical findings.

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.

Singularly perturbed reaction diffusion problem with large spatial delay via non-standard fitted operator method

In this article, a numerical method for singularly perturbed partial differential equations of the reaction-diffusion type with a large spatial delay is constructed. The method is developed using the Crank-Nicolson method for time derivative and a non-standard finite difference method for spatial derivative on a uniform mesh. Due to the presence of a small perturbation parameter, which multiplies the highest order derivative term, the solution exhibits twin boundary layers. The solution to the problem also shows an interior layer as a result of the delay in the spatial variable. To enhance the order of convergence, the Richardson extrapolation technique is applied. The convergence analysis of the proposed scheme is given. It is proved that the resulting scheme is uniformly convergent of fourth-order accurate in both space and time variables. Numerical experiments are conducted, and the results are in agreement with the theoretical findings. In addition, comparisons are performed, and the results show that the proposed scheme gives more accurate solutions and a higher rate of convergence than previous findings in the literature.

Numerical treatment of singularly perturbed unsteady Burger-Huxley equation

This article deals with the numerical treatment of a singularly perturbed unsteady Burger-Huxley equation. The equation is linearized using the Newton-Raphson-Kantorovich approximation method. The resulting linear equation is discretized using the implicit Euler method and an exponential spline method for time and space variables, respectively. Richardson's extrapolation technique is employed to increase the accuracy in the temporal direction. The stability and uniform convergence of the proposed scheme are investigated. The scheme is shown uniformly convergent with the order of convergence O(τ + ℓ2) and O(τ2 + ℓ2) before and after Richardson extrapolation, respectively. Several test examples are considered to validate the applicability and efficiency of the scheme. It is observed that the proposed scheme provides more accurate results than the methods available in the literature.

An optimal fitted numerical scheme for solving singularly perturbed parabolic problems with large negative shift and integral boundary condition

This paper deals with numerical solution of singularly perturbed parabolic partial differential equations with large negative shift on the spatial variable and integral boundary condition on the right side of the domain. For small values of perturbation parameter ɛ, solution of the problem exhibits an interior layer besides a parabolic boundary layers. The numerical methods combine the Crank–Nicolson difference scheme for the temporal direction on the uniform mesh and exponentially fitted operator finite difference method in spatial discretization. The integral boundary condition is treated using numerical integration techniques. The rate of convergence of the scheme is optimized by applying Richardson extrapolation technique. The stability and uniform convergence analysis has been carried out. To validate the applicability of the scheme, numerical examples are presented and the method is more accurate and shown to be second order uniformly convergent in both direction, and it also improves the results of the methods (Elango et al., 2021; Gobena and Duressa, 2021) existing in the literature.

A Nonstandard Fitted Operator Method for Singularly Perturbed Parabolic Reaction-Diffusion Problems with a Large Time Delay

In this paper, we design and investigate a higher order ε -uniformly convergent method to solve singularly perturbed parabolic reaction-diffusion problems with a large time delay. We use the Crank–Nicolson method for the time derivative, while the spatial derivative is discretized using a nonstandard finite difference approach on a uniform mesh. Furthermore, to improve the order of convergence, we used the Richardson extrapolation technique. The designed scheme converges independent of the perturbation parameter ( ε -uniformly convergent) and also achieves fourth-order convergent in both time and spatial variables. Two model examples are considered to demonstrate the applicability of the suggested method. The proposed method produces better accuracy and a higher rate of convergence than some methods that appear in the literature.

A Fitted Numerical Method for Interior-Layer Parabolic Convection–Diffusion Problems

The aim of this paper is to construct and analyze a fitted mesh finite difference method (FMFDM) for a class of time-dependent singularly perturbed convection–diffusion–reaction problems with a turning point. While such problems have been widely studied in the case of boundary layers, little has been achieved for interior layer problems. In addition, most of works have considered the coefficient functions as the functions of space only. In this work, we study problems where the coefficient functions are dependent on both space and time variables, and their solutions exhibit an interior layer. We derive the solution bounds and their derivatives. Then we use the classical implicit Euler method to discretize the time variable with a constant step-size. This method consists of two upwind finite difference schemes defined on an appropriate non-uniform mesh of Shishkin type. This mesh is fine near the turning point and coarse elsewhere. We prove that the method is almost first-order uniformly convergent with respect to the perturbation parameter. To improve the accuracy of the proposed method, we apply Richardson extrapolation technique. Numerical simulations are provided to confirm the theoretical findings.

Linear multistep methods and global Richardson extrapolation

In this work, we study the application the classical Richardson extrapolation (RE) technique to accelerate the convergence of sequences resulting from linear multistep methods (LMMs) for solving initial-value problems of systems of ordinary diffe- rential equations numerically. The advantage of the LMM-RE approach is that the combined method possesses higher order and favorable linear stability properties in terms of A- or A(α)-stability, and existing LMM codes can be used without any modification.

A high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution

The purpose of this paper is to introduce a high order convergent numerical method for singularly perturbed time dependent problems using mesh equidistribution. The discretization is based on the backward Euler scheme in time and a high order non-monotone scheme in space. In time direction we consider a uniform mesh, while in spatial direction we construct an adaptive mesh through equidistribution of a monitor function involving appropriate power of the solution’s second derivative. The method is analysed in two steps, splitting the time and space discretization errors. We establish that the method is uniformly convergent with optimal order having order one in time and order four in space. Further, we use the Richardson extrapolation technique for improving the order of convergence from one to two in time. Numerical experiments are presented to confirm the theoretically proven convergence result.