Partitioning methods for reaction–diffusion problems (original) (raw)
Related papers
Journal of Computational and Applied Mathematics, 2013
The numerical solution of a nonlinear degenerate reaction-diffusion equation of the quenching type is investigated. While spatial derivatives are discretized over symmetric nonuniform meshes, a Peaceman-Rachford splitting formula is employed to advance solutions of the semidiscretized system. The temporal step is determined adaptively through a suitable arc-length monitor function. A criteria is determined that guarantees that the numerical solution acquired preserves the positivity and monotonicity of the analytical solution. Weak stability is proven in a Von Neumann sense via the ∞-norm. Computational examples are given to illustrate our results.
2014
An enriched partition-of-unity (PU) finite element method is developed to solve timedependent diffusion problems. In the present PU formulation, an exponential solution describing the spatial diffusion decay is embedded in the finite element shape function. It results in an enriched approximation, which is in the form of local asymptotic expansion. The temporal decay in the solution is embedded naturally in the PU expansion so that, unlike previous works in this area, the same system matrices may be used for every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more robust and efficient, and yields more accurate solutions for a prescribed number of degrees of freedom. On the other hand, the notorious difficulty encountered in the meshless method in satisfying the essential boundary conditions is circumvented. Numerical results are presented for a transient diffusion equation with known analytical...
Application of a New Method of Discretization to Diffusion-Reaction Problems
International Journal of Modelling and Simulation, 2000
This paper presents a new integral solution method known as the Green Element Method (GEM). The technique of this solution, which relies essentially on the singular integral theory of the Boundary Element Method (BEM), is applied to problems involving mass diffusion and reaction in one dimension. This procedure employs the fundamental solution of the term with the highest derivative to construct a system of equations where both the dependent variable and its gradient become the primary variables. A linear weighting function in spatial dimensions is employed to approximate the scalar variables over the problem domain. The resulting banded coefficient matrix can be handled efficiently by matrix subroutines. Numerical results from test problems involving both linear and nonlinear reaction kinetics and different time discretization schemes show that this method is reliable and can produce accurate results.
A reaction pathway for a classical two-species reaction is considered with one reaction that is several orders of magnitudes faster than the other. To sustain the fast reaction, the transport and reaction effects must balance in such a way as to give an internal layer in space. For the steady-state problem, existing singular perturbation analysis rigorously proves the correct scaling of the internal layer. This work reports the results of exploratory numerical simulations that are designed to provide guidance for the analysis to be performed for the transient problem. The full model is comprised of a system of time-dependent reactiondiffusion equations coupled through the non-linear reaction terms with mixed Dirichlet and Neumann boundary conditions. In addition to internal layers in space, the time-dependent problem possesses an initial transient layer in time. To resolve both types of layers as accurately as possible, we design a finite element method with analytic evaluation of all integrals. This avoids all errors associated with the evaluation of the non-linearities and allows us to provide an analytic Jacobian matrix to the implicit time stepping method in the software package MATLAB. The simulation results show that the method resolves the localized sharp gradients accurately and can predict the scaling of the internal layers for the time-dependent problem. A comparison between our code and the established finite element package FEMLAB confirms the accuracy of our code. It also illustrates that our specialized implementation solves the problem significantly faster and requires substantially less memory.
Numerical methods for stiff reaction-diffusion systems, Discrete Contin
2007
Contributed in honor of Fred Wan on the occasion of his 70th birthday Abstract. In a previous study [21], a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has ex-cellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly un-conditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their ac-curacy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time steps. IIF2 is found to be more accurate for systems with large diffusion while...
A partition of unity FEM for time-dependent diffusion problems using multiple enrichment functions
International Journal for Numerical Methods in Engineering, 2013
An enriched partition of unity finite element method is developed to solve time dependent diffusion problems. In the present formulation, multiple exponential functions describing the spatial and temporal diffusion decay are embedded in the finite element approximation space. The resulting enrichment is in the form of a local asymptotic expansion. Unlike previous works in this area where the enrichment must be updated at each time step, here the temporal decay in the solution are embedded in the asymptotic expansion. Thus the system matrix which is evaluated at the first time step may be decomposed and retained for every next time step by just updating the right hand side of the linear system of equations. The advantage is a significant saving in the computational effort where previously the linear system must be reevaluated and resolved at every time step. In comparison with the traditional finite element analysis with p-version refinements, the present approach is much simpler, more efficient, and yields more accurate solutions for a prescribed number of degrees of freedom. Numerical results are presented for a transient diffusion equation with known analytical solution. The performance of the method is analyzed on two applications: the transient heat equation with a single source and with multiple sources. The aim of such a method compared to the classical finite element method is to solve time dependent diffusion applications efficiently and with an appropriate level of accuracy.
Numerical methods for stiff reaction-diffusion systems
Discrete and Continuous Dynamical Systems - Series B, 2007
In a previous study , a class of efficient semi-implicit schemes was developed for stiff reaction-diffusion systems. This method which treats linear diffusion terms exactly and nonlinear reaction terms implicitly has excellent stability properties, and its second-order version, with a name IIF2, is linearly unconditionally stable. In this paper, we present another linearly unconditionally stable method that approximates both diffusions and reactions implicitly using a second order Crank-Nicholson scheme. The nonlinear system resulted from the implicit approximation at each time step is solved using a multi-grid method. We compare this method (CN-MG) with IIF2 for their accuracy and efficiency. Numerical simulations demonstrate that both methods are accurate and robust with convergence using even very large size of time step. IIF2 is found to be more accurate for systems with large diffusion while CN-MG is more efficient when the number of spatial grid points is large.
Matrix-oriented discretization methods for reaction–diffusion PDEs: Comparisons and applications
Computers & Mathematics with Applications, 2020
Systems of reaction-diffusion partial differential equations (RD-PDEs) are widely applied for modelling life science and physico-chemical phenomena. In particular, the coupling between diffusion and nonlinear kinetics can lead to the so-called Turing instability, giving rise to a variety of spatial patterns (like labyrinths, spots, stripes, etc.) attained as steady state solutions for large time intervals. To capture the morphological peculiarities of the pattern itself, a very fine space discretization may be required, limiting the use of standard (vector-based) ODE solvers in time because of excessive computational costs. We show that the structure of the diffusion matrix can be exploited so as to use matrix-based versions of time integrators, such as Implicit-Explicit (IMEX) and exponential schemes. This implementation entails the solution of a sequence of discrete matrix problems of significantly smaller dimensions than in the vector case, thus allowing for a much finer problem discretization. We illustrate our findings by numerically solving the Schnackenberg model, prototype of RD-PDE systems with Turing pattern solutions, and the DIB-morphochemical model describing metal growth during battery charging processes.
2009
A new family of linearly implicit fractional step methods is proposed for the efficient numerical solution of a class of nonlinear time-dependent reaction-diffusion equations. By using the method of lines, the original problem is first discretized in space via a mimetic finite difference technique. The resulting differential system of stiff nonlinear equations is locally decomposed by suitable Taylor expansions and a domain decomposition splitting for the linear terms. This splitting is then combined with a linearly implicit one-step scheme belonging to the class of so-called fractional step Runge-Kutta methods. In this way, the original problem is reduced to the solution of several linear systems per time step which can be trivially decomposed into a set of uncoupled subsystems. As compared to classical domain decomposition techniques, our proposal does not require any Schwarz iterative procedure. The convergence of the designed method is illustrated by numerical experiments.
A posteriori analysis of an iterative multi-discretization method for reaction–diffusion systems
Computer Methods in Applied Mechanics and Engineering, 2013
This paper is concerned with the accurate computational error estimation of numerical solutions of multi-scale, multi-physics systems of reaction-diffusion equations. Such systems can present significantly different temporal and spatial scales within the components of the model, indicating the use of independent discretizations for different components. However, multi-discretization can have significant effects on accuracy and stability. We perform an adjoint-based analysis to derive asymptotically accurate a posteriori error estimates for a user-defined quantity of interest. These estimates account for leading order contributions to the error arising from numerical solution of each component, an error due to incomplete iteration, an error due to linearization, and for errors arising due to the projection of solution components between different spatial meshes. Several numerical examples with various settings are given to demonstrate the performance of the error estimators.