A finite element method adaptive in space and time for nonlinear reaction-diffusion systems

1992 ◽  
Vol 4 (4) ◽  
pp. 269-314 ◽  
Author(s):  
Jens Lang ◽  
Artur Walter
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Reaction Diffusion Systems ◽  
Space And Time ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Element Method

Finite element modeling of nonlinear reaction–diffusion–advection systems of equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2688-2715 ◽  
Author(s):  
Sanjay Komala Sheshachala ◽  
Ramon Codina
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Reaction Diffusion ◽  
Diffusion Reaction ◽  
Content Type ◽  
Numerical Examples ◽  
Stabilized Finite Element ◽  
Stabilized Finite Element Method ◽  
Nonlinear Reaction ◽  
Element Method

Purpose This paper aims to present a finite element formulation to approximate systems of reaction–diffusion–advection equations, focusing on cases with nonlinear reaction. The formulation is based on the orthogonal sub-grid scale approach, with some simplifications that allow one to stabilize only the convective term, which is the source of potential instabilities. The space approximation is combined with finite difference time integration and a Newton–Raphson linearization of the reactive term. Some numerical examples show the accuracy of the resulting formulation. Applications using classical nonlinear reaction models in population dynamics are also provided, showing the robustness of the approach proposed. Design/methodology/approach A stabilized finite element method for advection–diffusion–reaction equations to the problem on nonlinear reaction is adapted. The formulation designed has been implemented in a computer code. Numerical examples are run to show the accuracy and robustness of the formulation. Findings The stabilized finite element method from which the authors depart can be adapted to problems with nonlinear reaction. The resulting method is very robust and accurate. The framework developed is applicable to several problems of interest by themselves, such as the predator–prey model. Originality/value A stabilized finite element method to problems with nonlinear reaction has been extended. Original contributions are the design of the stabilization parameters and the linearization of the problem. The application examples, apart from demonstrating the validity of the numerical model, help to get insight in the system of nonlinear equations being solved.

scholarly journals Numerical study of Fisher's equation by a Petrov-Galerkin finite element method

1991 ◽  
Vol 33 (1) ◽  
pp. 27-38 ◽  
Author(s):  
S. Tang ◽  
R. O. Weber
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Study ◽  
Galerkin Finite Element Method ◽  
Linear Diffusion ◽  
Galerkin Finite Element ◽  
Fisher’S Equation ◽  
Nonlinear Reaction ◽  
Fisher's Equation ◽  
Element Method

AbstractFisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by a Petrov-Galerkin finite element method. The results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are determined by the system itself and are independent of the initial values. Comparing with other studies, the numerical scheme used in this paper is satisfactory with regard to its accuracy and stability. It has the advantage of being much more concise.

scholarly journals CAS WAVELETS METHOD TO SOLVE REACTION-DIFFUSION SYSTEM AND COMPARE IT WITH (G.F.E) METHOD

2020 ◽  
Vol 17 (35) ◽  
pp. 1110-1123
Author(s):  
Badran Jasim SALIM ◽  
Oday Ahmed JASIM
Keyword(s):  
Steady State ◽  
Finite Elements ◽  
Reaction Diffusion ◽  
Partial Differential ◽  
Reaction Diffusion Systems ◽  
Time Frequency ◽  
Nonlinear Reaction ◽  
Diffusion Systems ◽  
Cas Wavelets ◽  
Steady State Solutions

Wavelet analysis plays a prominent role in various fields of scientific disciplines. Mainly, wavelets are very successfully used in signal analysis for waveform representation and segmentation, time-frequency analysis, and fast algorithms in the propagation equations and reaction. This research aimed to guide researchers to use Cos and Sin (CAS) to approximate the solution of the partial differential equation system. This method has been successfully applied to solve a coupled system of nonlinear Reaction-diffusion systems. It has been shown CAS wavelet method is quite capable and suited for finding exact solutions once the consistency of the method gives wider applicability where the main idea is to transform complex nonlinear partial differential equations into algebraic equation systems, which are easy to handle and find a numerical solution for them. By comparing the numerical solutions of the CAS and Galerkin finite elements methods, the answer of nonlinear Reaction-diffusion systems using the CAS wavelets for all tˆ and x values is accurate, reliable, robust, promising, and quickly arrives at the exact solution. When parameters 𝜀1 𝑎𝑛𝑑 𝜀2 are growing and with L decreasing, then the CAS method converges to steady-state solutions quickly (the less L, the more accurate the solution). It is converging towards steady-state solutions faster than and loses steps over time. Moreover, the results also show that the solution of the CAS wavelets is more reliable and faster compared to the Galerkin finite elements (G.F.E).

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