Solving reaction-diffusion equations 10 times faster

The most popular numerical method for solving systems of reaction-diffusion equations continues to be a low order finite-difference scheme coupled with low order Euler time stepping. This paper extends previous 1D work and reports experiments that show that with high--order methods one can speed up such simulations for 2D and 3D problems by factors of 10--100. A short MATLAB code (2/3D) that can serve as a template is included.\ud \ud This work was supported by the Engineering and Physical Sciences Research Council (UK) and by the MathWorks, Inc

Engaging Transformation: Using Seasonal Rounds to Anticipate Climate Change

Seasonal rounds are deliberative articulations of a community’s sociocultural relations with their ecological system. The process of visualizing seasonal rounds informs transdisciplinary research. We present a methodological approach for communities of enquiry to engage communities of practice through context-specific sociocultural and ecological relations driven by seasonal change. We first discuss historical précis of the concept of seasonal rounds that we apply to assess the spatial and temporal communal migrations and then describe current international research among Indigenous and rural communities in North America and Central Asia by the creation of a common vocabulary through mutual respect for multiple ways of knowing, validation of co-generated knowledge, and insights into seasonal change. By investigating the relationship between specific biophysical indicators and livelihoods of local communities, we demonstrate that seasonal rounds are an inclusive and participatory methodology that brings together diverse Indigenous and rural voices to anticipate anthropogenic climate change

High order timestepping for stiff semilinear partial differential equations

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Fourth-order time stepping for stiff PDEs

A modification of the ETDRK4 (Exponential Time Differencing fourth-order Runge-Kutta) method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to non-diagonal operators. A comparison is made of the performance of this modified ETD scheme against the competing methods of implicit--explicit differencing, integrating factors, time-splitting, and Fornberg and Driscoll's "sliders" for the KdV, Kuramoto-Sivashinsky, Burgers, and Allen-Cahn equations in one space dimension. Implementation of the method is illustrated by short MATLAB programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best

Fourth-order time stepping for stiff PDEs

Abstract. A modification of the exponential time-differencing fourth-order Runge–Kutta method for solving stiff nonlinear PDEs is presented that solves the problem of numerical instability in the scheme as proposed by Cox and Matthews and generalizes the method to nondiagonal operators. A comparison is made of the performance of this modified exponential time-differencing (ETD) scheme against the competing methods of implicit-explicit differencing, integrating factors, time-splitting, and Fornberg and Driscoll’s “sliders ” for the KdV, Kuramoto–Sivashinsky, Burgers, and Allen–Cahn equations in one space dimension. Implementation of the method is illustrated by short Matlab programs for two of the equations. It is found that for these applications with fixed time steps, the modified ETD scheme is the best