scholarly journals Wavelet Method for Numerical Solution of Parabolic Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
A. H. Choudhury
Keyword(s):  
Parabolic Equations ◽  
Space Dimension ◽  
Neumann Boundary ◽  
Time Variable ◽  
Basis Functions ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Parabolic Partial Differential Equations ◽  
Convection Diffusion ◽  
Diffusion And Convection

We derive a highly accurate numerical method for the solution of parabolic partial differential equations in one space dimension using semidiscrete approximations. The space direction is discretized by wavelet-Galerkin method using some special types of basis functions obtained by integrating Daubechies functions which are compactly supported and differentiable. The time variable is discretized by using various classical finite difference schemes. Theoretical and numerical results are obtained for problems of diffusion, diffusion-reaction, convection-diffusion, and convection-diffusion-reaction with Dirichlet, mixed, and Neumann boundary conditions. The computed solutions are highly favourable as compared to the exact solutions.

scholarly journals Well-Posedness and Blow Up for IBVP for Semilinear Parabolic Equations and Numerical Methods

2010 ◽  
Vol 10 (4) ◽  
pp. 395-421 ◽  
Author(s):  
P. Matus ◽  
S. Lemeshevsky ◽  
A. Kandratsiuk
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Nonlinear Differential Equations ◽  
Finite Difference Schemes ◽  
Differential Problem ◽  
Nonlinear Convection ◽  
Convection Diffusion ◽  
Convection Diffusion Equation ◽  
The Stability ◽  
Semilinear Parabolic

Abstract We have studied the stability of finite-difference schemes approximating boundary value problems for parabolic equations with a nonlinear and nonmonotonic source of the power type. We have obtained simple sufficient input data conditions, in which the solution of the differential problem is globally stable for all 0 ≤ t ≤ +∞. It is shown that if these conditions fail, then the solution can blow up (go to infinity) in finite time. The lower bound of the blow up time has been determined. The stability of the solution of BVP for the nonlinear convection-diffusion equation has been investigated. In all cases, we used the method of energy inequalities based on the application of the Chaplygin comparison theorem for nonlinear differential equations, Bihari-type inequalities and their discrete analogs.

scholarly journals L2-stability of a finite element – finite volume discretization of convection-diffusion-reaction equations with nonhomogeneous mixed boundary conditions

2017 ◽  
Vol 51 (3) ◽  
pp. 919-947 ◽  
Author(s):  
Paul Deuring ◽  
Robert Eymard
Keyword(s):  
Finite Element ◽  
Boundary Conditions ◽  
Finite Volume ◽  
Mixed Boundary ◽  
Neumann Boundary ◽  
Diffusion Reaction ◽  
Triangular Grid ◽  
Diffusion Term ◽  
Negative Power ◽  
Convection Diffusion

We consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin−Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discretized by Crouzeix−Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In the unsteady case, the implicit Euler method is used as time discretization. This scheme is shown to be unconditionally L2-stable, uniformly with respect to diffusion, except if the Robin−Neumann boundary condition is inhomogeneous and the convective velocity is tangential at some points of the Robin−Neumann boundary. In that case, a negative power of the diffusion coefficient arises. As is shown by a counterexample, this exception cannot be avoided.

Essentially non-oscillatory and weighted essentially non-oscillatory schemes

Acta Numerica ◽  
2020 ◽  
Vol 29 ◽  
pp. 701-762
Author(s):  
Chi-Wang Shu
Keyword(s):  
Main Idea ◽  
High Order Accuracy ◽  
Finite Difference Schemes ◽  
Approximation Procedure ◽  
Discontinuous Solutions ◽  
Weno Schemes ◽  
Convection Diffusion ◽  
Different Types ◽  
Basic Ideas ◽  
Weighted Eno

Essentially non-oscillatory (ENO) and weighted ENO (WENO) schemes were designed for solving hyperbolic and convection–diffusion equations with possibly discontinuous solutions or solutions with sharp gradient regions. The main idea of ENO and WENO schemes is actually an approximation procedure, aimed at achieving arbitrarily high-order accuracy in smooth regions and resolving shocks or other discontinuities sharply and in an essentially non-oscillatory fashion. Both finite volume and finite difference schemes have been designed using the ENO or WENO procedure, and these schemes are very popular in applications, most noticeably in computational fluid dynamics but also in other areas of computational physics and engineering. Since the main idea of the ENO and WENO schemes is an approximation procedure not directly related to partial differential equations (PDEs), ENO and WENO schemes also have non-PDE applications. In this paper we will survey the basic ideas behind ENO and WENO schemes, discuss their properties, and present examples of their applications to different types of PDEs as well as to non-PDE problems.

scholarly journals Reliable Efficient Difference Methods for Random Heterogeneous Diffusion Reaction Models with a Finite Degree of Randomness

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 206
Author(s):  
María Consuelo Casabán ◽  
Rafael Company ◽  
Lucas Jódar
Keyword(s):  
Stochastic Process ◽  
Finite Difference ◽  
Coming Out ◽  
Finite Difference Methods ◽  
Diffusion Reaction ◽  
Finite Difference Schemes ◽  
Finite Degree ◽  
Mean Square Stability ◽  
Reaction Models ◽  
Difference Methods

This paper deals with the search for reliable efficient finite difference methods for the numerical solution of random heterogeneous diffusion reaction models with a finite degree of randomness. Efficiency appeals to the computational challenge in the random framework that requires not only the approximating stochastic process solution but also its expectation and variance. After studying positivity and conditional random mean square stability, the computation of the expectation and variance of the approximating stochastic process is not performed directly but through using a set of sampling finite difference schemes coming out by taking realizations of the random scheme and using Monte Carlo technique. Thus, the storage accumulation of symbolic expressions collapsing the approach is avoided keeping reliability. Results are simulated and a procedure for the numerical computation is given.

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