Journal of Computational Physics
Parallel, adaptive finite element methods for conservation laws
Proceedings of the third ARO workshop on Adaptive methods for partial differential equations
Journal of Computational Physics
The Runge-Kutta discontinuous Galerkin method for conservation laws V multidimensional systems
Journal of Computational Physics
Trapezoidal and midpoint splittings for initial-boundary value problems
Mathematics of Computation
A discontinuous hp finite element method for diffusion problems
Journal of Computational Physics
The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems
SIAM Journal on Numerical Analysis
Stiff differential equations solved by Radau methods
Proceedings of the on Numerical methods for differential equations
An analysis of operator splitting techniques in the stiff case
Journal of Computational Physics
Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems
Journal of Scientific Computing
A Local Discontinuous Galerkin Method for KdV Type Equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
High-Order WENO Schemes for Hamilton-Jacobi Equations on Triangular Meshes
SIAM Journal on Scientific Computing
Journal of Scientific Computing
A moving grid finite element method applied to a model biological pattern generator
Journal of Computational Physics
Explicit Time-Stepping for Stiff ODEs
SIAM Journal on Scientific Computing
A comparison of implicit time integration methods for nonlinear relaxation and diffusion
Journal of Computational Physics
Local discontinuous Galerkin methods for nonlinear dispersive equations
Journal of Computational Physics
Studies of the accuracy of time integration methods for reaction-diffusion equations
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Adaptive moving mesh computations for reaction--diffusion systems
Journal of Computational and Applied Mathematics - Special issue: Selected papers from the 2nd international conference on advanced computational methods in engineering (ACOMEN2002) Liege University, Belgium, 27-31 May 2002
RKC time-stepping for advection-diffusion-reaction problems
Journal of Computational Physics
Journal of Scientific Computing
Local discontinuous Galerkin methods for nonlinear Schrödinger equations
Journal of Computational Physics
High Order A-stable Numerical Methods for Stiff Problems
Journal of Scientific Computing
Journal of Computational Physics
Journal of Computational Physics
Journal of Computational Physics
Journal of Scientific Computing
Two-Grid Method for Nonlinear Reaction-Diffusion Equations by Mixed Finite Element Methods
Journal of Scientific Computing
The finite volume spectral element method to solve Turing models in the biological pattern formation
Computers & Mathematics with Applications
Finite volume element approximation of an inhomogeneous Brusselator model with cross-diffusion
Journal of Computational Physics
Hi-index | 0.01 |
Nonlinear reaction-diffusion systems which are often employed in mathematical modeling in developmental biology are usually highly stiff in both diffusion and reaction terms. Moreover, they are typically considered on multidimensional complex geometrical domains because of complex shapes of embryos. We overcome these computational challenges by combining discontinuous Galerkin (DG) finite element methods with Strang type symmetrical operator splitting technique, on triangular meshes. This allows us to avoid directly solving a coupled nonlinear system, as is necessary with the standard implicit schemes. Numerical solutions of two reaction-diffusion systems, the well-studied Schnakenberg model, which has been applied to several problems in developmental biology, and a new biologically based system for skeletal pattern formation in the vertebrate limb, are presented to demonstrate effects of various domain geometries on the resulting biological patterns.