Temporal derivatives in the finite-element method on continuously deforming grids
SIAM Journal on Numerical Analysis
Implicit-explicit methods for time-dependent partial differential equations
SIAM Journal on Numerical Analysis
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
Fourth-Order Time-Stepping for Stiff PDEs
SIAM Journal on Scientific Computing
Journal of Scientific Computing
Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains
Journal of Computational Physics
Second-order implicit-explicit scheme for the Gray-Scott model
Journal of Computational and Applied Mathematics
Computers in Biology and Medicine
Journal of Scientific Computing
An efficient and robust numerical algorithm for estimating parameters in Turing systems
Journal of Computational Physics
Computer Methods and Programs in Biomedicine
Numerical approximation of Turing patterns in electrodeposition by ADI methods
Journal of Computational and Applied Mathematics
Journal of Computational Physics
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In this paper, we illustrate the application of time-stepping schemes to reaction-diffusion systems on fixed and continuously growing domains by use of finite element and moving grid finite element methods. We present two schemes for our studies, namely a first-order backward Euler finite differentiation formula coupled with a special form of linearisation of the nonlinear reaction terms (1-SBEM) and a second-order semi-implicit backward finite differentiation formula (2-SBDF) with no linearisation of the reaction terms. Our results conclude that for the type of reaction-diffusion systems considered in this paper, the 1-SBEM is more stable than the 2-SBDF scheme and that the 1-SBEM scheme has a larger region of stability (at least by a factor of 10) than that of the 2-SBDF scheme. As a result, the 1-SBEM scheme becomes a natural choice when solving reaction-diffusion problems on continuously deforming domains.