Numerical methods for stiff reaction-diffusion systems (original) (raw)
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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...
Efficient semi-implicit schemes for stiff systems
Journal of Computational Physics, 2006
When explicit time discretization schemes are applied to stiff reaction-diffusion equations, the stability constraint on the time step depends on two terms: the diffusion and the reaction. The part of the stability constraint due to diffusion can be totally removed if the linear diffusions are treated exactly using integration factor (IF) or exponential time differencing (ETD) methods. For systems with severely stiff reactions, those methods are not efficient because the reaction terms in IF or ETD are still approximated with explicit schemes. In this paper, we introduce a new class of semi-implicit schemes, which treats the linear diffusions exactly and explicitly, and the nonlinear reactions implicitly. A distinctive feature of the scheme is the decoupling between the exact evaluation of the diffusion terms and implicit treatment of the nonlinear reaction terms. As a result, the size of the nonlinear system arising from the implicit treatment of the reactions is independent of the number of spatial grid points; it only depends on the number of original equations, unlike the case in which standard implicit temporal schemes are directly applied to the reaction-diffusion system. The stability region for this class of methods is much larger than existing methods using an explicit treatment of reaction terms. In particular, the one with second order accuracy is unconditionally linearly stable with respect to both diffusion and reaction. Direct numerical simulations on test equations, as well as morphogen systems from developmental biology, show the new semi-implicit schemes are efficient, robust and accurate. (Q. Nie).
Finite Difference Implicit Schemes to Coupled Two-Dimension Reaction Diffusion System
Journal of Applied Mathematics and Physics
In this research article, two finite difference implicit numerical schemes are described to approximate the numerical solution of the two-dimension modified reaction diffusion Fisher's system which exists in coupled form. Finite difference implicit schemes show unconditionally stable and second-order accurate nature of computational algorithm also the validation and comparison of analytical solution, are done through the examples having known analytical solution. It is found that the numerical schemes are in excellent agreement with the analytical solution. We found, second-implicit scheme is much faster than the first with good rate of convergence also we used NVIDA devices to accelerate the computations and efficiency of the algorithm. Numerical results show our proposed schemes with use of HPC (High performance computing) are very efficient and reliable.
Robust IMEX Schemes for Solving Two-Dimensional Reaction–Diffusion Models
International Journal of Nonlinear Sciences and Numerical Simulation, 2015
In this paper, numerical simulations of two-dimensional reaction–diffusion (for single and multi-species) models are considered for pattern formation processes. The nature of our problems permits the use of two classical approaches. These semi-linear partial differential equations are split into a linear equation which contains the highly stiff part of the problem, and a nonlinear part that is expected to be varying slowly than the linear part. For the spatial discretization, we introduce higher-order symmetric finite difference scheme, and the resulting ordinary differential equations are then solved with the use of the family of implicit–explicit (IMEX) schemes. Stability properties of these schemes as well as the linear stability analysis of the problems are well presented. Numerical examples and results are also given to illustrate the accuracy and implementation of the methods.
A Semi-implicit Numerical Scheme for Reacting Flow
Journal of Computational Physics, 1998
A stiff, 1 operator-split projection scheme is constructed for the simulation of unsteady two-dimensional reacting flow with detailed kinetics. The scheme is based on the compressible conservation equations for an ideal gas mixture in the zero-Machnumber limit. The equations of motion are spatially discretized using second-order centered differences and are advanced in time using a new stiff predictor-corrector approach. The new scheme is a modified version of the additive, stiff scheme introduced in a previous effort by H. N. Najm, P. S. Wyckoff, and O. M. Knio (1998, J. Comput. Phys. 143, 381). The predictor updates the scalar fields using a Strang-type operator-split integration step which combines several explicit diffusion sub-steps with a single stiff step for the reaction terms, such that the global time step may significantly exceed the critical diffusion stability limit. Convection terms are explicitly handled using a second-order multi-step scheme. The velocity field is advanced using a projection scheme which consists of a partial convection-diffusion update followed by a pressure correction step. A split approach is also adopted for the convection-diffusion step in the momentum update. This splitting combines an explicit treatment of the convective terms at the global time step with several explicit fractional steps for diffusion. Finally, a corrector step is implemented in order to couple the evolution of the density and velocity fields and to stabilize the computations. The corrector acts only on the convective terms and the pressure field, while the predicted updates due to diffusion and reaction are left unchanged. The correction of the scalar fields is implemented using a single-step non-split, non-stiff, second-order time integration. A similar procedure is used for the velocity field, which is followed OPERATOR-SPLIT, STIFF SCHEME 429 by a pressure projection step. The performance and behavior of the operator-split scheme are first analyzed based on tests for a nonlinear reaction-diffusion equation in one space dimension, followed by computations with a detailed C 1 C 2 methane-air mechanism in one and two dimensions. The tests are used to verify that the scheme is effectively second order in time, and to suggest guidelines for selecting integration parameters, including the number of fractional diffusion steps and tolerance levels in the stiff integration. For two-dimensional simulations with the present reaction mechanism, flame conditions, and resolution parameters, speedup factors of about 5 are achieved over the previous additive scheme, and about 25 over the original explicit scheme.
Numerical Approximation to Nonlinear One Dimensional Coupled Reaction Diffusion System
Journal of Applied Mathematics and Physics
This research paper represents a numerical approximation to non-linear coupled one dimension reaction diffusion system, which includes the existence and uniqueness of the time dependent solution with upper and lower bounds of the solution. Also numerical approximation is obtained by finite difference schemes to reach at reasonable level of accuracy, which is magnified by 2 L , L ∞ and relative error norms. The accuracy of the approximations is shown by randomly selected grid points along time level and comparison with analytical results. The test example demonstrates the accuracy, efficiency and versatility of the proposed schemes. Moreover, the schemes can be easily applied to a wide class of higher dimension non-linear reaction diffusion equations with a little modifications.
Journal of Computational Physics
Implicit integration factor (IIF) methods were developed in the literature for solving time-dependent stiff partial differential equations (PDEs). Recently, IIF methods are combined with weighted essentially non-oscillatory (WENO) schemes in [Jiang and Zhang, Journal of Computational Physics, 253 (2013) 368-388] to efficiently solve stiff nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the non-stiff hyperbolic part of the system. To efficiently calculate large matrix exponentials, Krylov subspace approximation is directly applied to the implicit integration factor (IIF) methods. So far, the IIF methods developed in the literature are multistep methods. In this paper, we develop Krylov single-step IIF-WENO methods for solving stiff advection-diffusionreaction equations. The methods are designed carefully to avoid generating positive exponentials in the matrix exponentials, which is necessary for the stability of the schemes. We analyze the stability and truncation errors of the single-step IIF schemes. Numerical examples of both scalar equations and systems are shown to demonstrate the accuracy, efficiency and robustness of the new methods.
Numerical study to coupled three dimensional reaction diffusion system
IEEE Access
The numerical solution of reaction diffusion systems may require more computational efforts if the change in concentrations occurs extremely rapid. This is because more time points are needed to resolve the reaction diffusion process accurately. In this article three finite difference implicit schemes are used which are unconditionally stable in order to enhance consistency. Novelty is reported by compact finite difference implicit scheme on reaction diffusion system with higher accuracy measured by L 2 , L∞ & Relativeerror norms. Efficiency is observed by reducing grid space along small temporal steps. CPU performance, transmission capacity along comparison of three schemes shows excellent agreement with analytical solution.
Journal of Computational Physics, 2013
Implicit integration factor (IIF) methods are originally a class of efficient "exactly linear part" time discretization methods for solving time-dependent partial differential equations (PDEs) with linear high order terms and stiff lower order nonlinear terms. For complex systems (e.g. advection-diffusion-reaction (ADR) systems), the highest order derivative term can be nonlinear, and nonlinear nonstiff terms and nonlinear stiff terms are often mixed together. High order weighted essentially non-oscillatory (WENO) methods are often used to discretize the hyperbolic part in ADR systems. There are two open problems on IIF methods for solving ADR systems: (1) how to obtain higher than the second order global time discretization accuracy; (2) how to design IIF methods for solving fully nonlinear PDEs, i.e., the highest order terms are nonlinear. In this paper, we solve these two problems by developing new Krylov IIF-WENO methods to deal with both semilinear and fully nonlinear advection-diffusion-reaction equations. The methods can be designed for arbitrary order of accuracy. The stiffness of the system is resolved well and the methods are stable by using time step sizes which are just determined by the nonstiff hyperbolic part of the system. Large time step size computations are obtained. We analyze the stability and truncation errors of the schemes. Numerical examples of both scalar equations and systems in two and three spatial dimensions are shown to demonstrate the accuracy, efficiency and robustness of the methods.
Linearly implicit Runge–Kutta methods for advection–reaction–diffusion equations
Applied Numerical Mathematics, 2001
We construct two variable-step linearly implicit Runge-Kutta methods of orders 3 and 4 for the numerical integration of the semidiscrete equations arising after the spatial discretization of advection-reaction-diffusion equations. We study the stability properties of these methods giving the appropriate extension of the concept of L-stability. Numerical results are reported when the methods presented are combined with spectral discretizations. Our experiments show that the methods, being easily implementable, can be competitive with standard stiffly accurate time integrators.