Abstract
We present a new numerical method for solving nonlinear reaction-diffusion systems with cross-diffusion which are often taken as mathematical models for many applications in the biological, physical, and chemical sciences. The two-dimensional system is discretized by the local discontinuous Galerkin (LDG) method on unstructured triangular meshes associated with the piecewise linear finite element spaces, which can derive not only numerical solutions but also approximations for fluxes at the same time comparing with most of study work up to now which has derived numerical solutions only. Considering the stability requirement for the explicit scheme with strict time step restriction (Δt=O(hmin2)), the implicit integration factor (IIF) method is employed for the temporal discretization so that the time step can be relaxed asΔt=O(hmin). Moreover, the method allows us to compute element by element and avoids solving a global system of nonlinear algebraic equations as the standard implicit schemes do, which can reduce the computational cost greatly. Numerical simulations about the system with exact solution and the Brusselator model, which is a theoretical model for a type of autocatalytic chemical reaction, are conducted to confirm the expected accuracy, efficiency, and advantages of the proposed schemes.
Highlights
In 1952, Turing proposed the reaction-diffusion systems in the seminal paper [1], which constitute an essential basis to describe morphogenetic mechanisms
We present the fully discrete scheme, which was obtained by combining the local discontinuous Galerkin (LDG) method with the integration factor (IIF) method, to solve the nonlinear reaction-diffusion system (1)
From the numerical simulations in Example 3, we have found that the patterns of numerical solutions uh and Vh are always of the same type
Summary
In 1952, Turing proposed the reaction-diffusion systems in the seminal paper [1], which constitute an essential basis to describe morphogenetic mechanisms. In a reaction-diffusion system describing the interaction between two species, different diffusion rates can lead to the destabilization of a constant steady state, followed by the transition to a nonhomogeneous steady state According to this result, the equilibrium of the nonlinear system is asymptotically stable in the absence of diffusion but unstable in the presence of diffusion, which is called Turing unstable [2, 3]. We choose to pursue LDG method, where more general numerical fluxes than those in [26] are used, coupled with Krylov implicit integration factor (IIF) methods [30, 31] for temporal discretization which is based on the IIF method [32] By applying this method, we can derive the numerical approximations for solutions and for fluxes at the same time.
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