Six numerical schemes for parabolic initial boundary value problems with a priori bounded solution (original) (raw)

Solving second order non-linear parabolic PDEs using generalized finite difference method (GFDM)

Journal of Computational and Applied Mathematics, 2018

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): wave propagation, advection-diffusion, plates, beams, etc. The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear parabolic PDEs. This paper illustres that the GFD explicit formulae developed to obtain the different derivatives of the pde's are based in the existence of a positive definite matrix that it is obtained using moving least squares approximation and Taylor series development. Criteria for convergence of fully explicit method using GFDM for different non linear parabolic pdes are given. This paper shows the application of the GFDM to solving different non-linear problems including applications to heat transfer, acoustics and problems of mass transfer.

Application of the generalized finite difference method to solve the advection–diffusion equation

Journal of Computational and Applied Mathematics, 2011

The possibility of using a nodal method allowing irregular distribution of nodes in a natural way is one of the main advantages of the generalized finite difference method (GFDM) with regard to the classical finite difference method. Moreover, this feature has made it one of the most-promising meshless methods because it also allows us to reduce the time-consuming task of mesh generation and the numerical solution of integrals. This characteristic allows us to shape geological features easily whilst maintaining accuracy in the results, which can be a source of great interest when dealing with this kind of problems. Two widespread geophysical investigation methods in civil engineering are the cross-hole method and the seismic refraction method. This paper shows the use of the GFDM to model the aforementioned geophysical investigation tests showing precision in the obtained results when comparing them with experimental data.

Application of Some Finite Difference Schemes for Solving One Dimensional Diffusion Equation

American Scientific Research Journal for Engineering, Technology, and Sciences, 2016

In this paper the numerical solutions of one dimensional diffusion equation using some finite difference methods have been considered. For that purpose three examples of the diffusion equation together with different boundary conditions are examined. The finite difference methods applied on each example are (i) forward time centered space (ii) backward time centered space and (iii) Crank – Nicolson. In each case, we have studied stability of finite difference method and also obtained numerical result. The performance of each scheme is evaluated in accordance with both the accuracy of the solution and programming efforts. The implementation and behavior of the schemes have been compared and the results are illustrated pictorially. It is found in case of the test examples studied here that the Crank – Nicolson scheme gives better approximations than the two other schemes.

Finite difference scheme for a singularly perturbed parabolic equations in the presence of initial and boundary layers

Mathematical Modelling and Analysis, 2008

The grid approximation of an initial-boundary value problem is considered for a singularly perturbed parabolic reaction-diffusion equation. The secondorder spatial derivative and the temporal derivative in the differential equation are multiplied by parameters ε 2 1 and ε 2 2 , respectively, that take arbitrary values in the open-closed interval (0, 1]. The solutions of such parabolic problems typically have boundary, initial layers and/or initial-boundary layers. A priori estimates are constructed for the regular and singular components of the solution. Using such estimates and the condensing mesh technique for a tensor-product grid, piecewise-uniform in x and t, a difference scheme is constructed that converges ε-uniformly at the rate O(N −2 ln 2 N + N −1 0 ln N0), where (N + 1) and (N0 + 1) are the numbers of mesh points in x and t respectively.

Solving second order non-linear hyperbolic PDEs using generalized finite difference method (GFDM)

Journal of Computational and Applied Mathematics, 2020

The generalized finite difference method (GFDM) has been proved to be a good meshless method to solve several linear partial differential equations (PDEs): wave propagation, advection-diffusion, plates, beams, etc. The GFDM allows us to use irregular clouds of nodes that can be of interest for modelling non-linear hyperbolic PDEs. This paper illustrates that the GFD explicit formulae developed to obtain the different derivatives of the PDEs are based on the existence of a positive definite matrix which is obtained using moving least squares approximation and Taylor series development. Consistency and stability are shown in this paper for semilinear and quasilinear hyperbolic equations. This paper shows the application of the GFDM for solving second order non-linear hyperbolic problems.

A computational study of three numerical methods for some advection-diffusion problems

Applied Mathematics and Computation, 2016

Three numerical methods have been used to solve two problems described by advection-diffusion equations with specified initial and boundary conditions. The methods used are the third order upwind scheme [4], fourth order upwind scheme [4] and Non-Standard Finite Difference scheme (NSFD) [9]. We considered two test problems. The first test problem has steep boundary layers near x = 1 and this is challenging problem as many schemes are plagued by non-physical oscillation near steep boundaries [15]. Many methods suffer from computational noise when modelling the second test problem especially when the coefficient of diffusivity is very small for instance 0.01. We compute some errors, namely L 2 and L ∞ errors, dissipation and dispersion errors, total variation and the total mean square error for both problems and compare the computational time when the codes are run on a matlab platform. We then use the optimization technique devised by Appadu [1] to find the optimal value of the time step at a given value of the spatial step which minimizes the dispersion error and this is validated by some numerical experiments.

On Some Finite Difference Schemes for the Solutions of Parabolic Partial Differential Equations

2023

This paper presents the comparison of three different and unique finite difference schemes used for finding the solutions of parabolic partial differential equations (PPDE). Knowing fully that the efficiency of a numerical schemes depends solely on their stability therefore, the schemes were compared based on their stability using von Newmann method. The implicit scheme and Dufort-Frankel schemes using von Newmann stability method are unconditionally stable, while the explicit scheme is conditionally stable. The schemes were also applied to solve a one dimensional parabolic partial differential equations (heat equation) numerically and their results compared for best in efficiency. The numerical experiments as seen in the tables presented and also the percentage errors, which proves that the implicit scheme is good compare to the other two schemes. Also, the implementation of the implicit scheme is faster than that of the explicit and Dufort-Frankel schemes. The results obtained in work also compliment and agrees with the results in literature.

On Quarter-Sweep Finite Difference Scheme for One-Dimensional Porous Medium Equations

International Journal of Apllied Mathematics, 2020

In this article, we introduce an implicit finite difference approximation for one-dimensional porous medium equations using Quarter-Sweep approach. We approximate the solutions of the nonlinear porous medium equations by the application of the Newton method and use the Gauss-Seidel iteration. This yields a numerical method that reduces the computational complexity when the spatial grid spaces are reduced. The numerical result shows that the proposed method has a smaller number of iterations, a shorter computation time and a good accuracy compared to Newton-Gauss-Seidel and Half-Sweep Newton-Gauss-Seidel methods.

Fourth-order finite difference method for 2D parabolic partial differential equations with nonlinear first-derivative terms

Numerical Methods for Partial Differential Equations, 1992

We attempt to obtain a two-level implicit finite difference scheme using nine spatial grid points of O(k2 + kh2 + h4) for solving the 2D nonlinear parabolic partial differential equation v1uxx + v2uyy = f(x, y, t, u, ux, uy, u1) where v1 and v2 are positive constants, with Dirichlet boundary conditions. The method, when applied to a linear diffusion-convection problem, is shown to be unconditionally stable. Computational efficiency and the results of numerical experiments are discussed.