An adaptive splitting approach for the quenching solution of reaction–diffusion equations over nonuniform grids

Beauregard, Matthew A.; Sheng, Qin

doi:10.1016/j.cam.2012.10.005

Cited by 26 publications

(52 citation statements)

References 19 publications

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“…Since M i j ≤ 0 for i j and the weak row sum criterion is satisfied, we have that M −1 exists and that M is inverse positive [6,9]. A similar argument gives that I − τ k 2 R is nonsingular and inverse positive.…”

Section: Numerical Analysismentioning

confidence: 99%

A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion

Beauregard

Padgett

Parshad

2017

Journal of Computational and Applied Mathematics

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Systems of reaction-diffusion equations are commonly used in biological models of food chains. The populations and their complicated interactions present numerous challenges in theory and in numerical approximation. In particular, self-diffusion is a nonlinear term that models overcrowding of a particular species. The nonlinearity complicates attempts to construct efficient and accurate numerical approximations of the underlying systems of equations. In this paper, a new nonlinear splitting algorithm is designed for a partial differential equation that incorporates selfdiffusion. We present a general model that incorporates self-diffusion and develop a numerical approximation. The numerical analysis of the approximation provides criteria for stability and convergence. Numerical examples are used to illustrate the theoretical results.

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Section: Numerical Analysismentioning

confidence: 99%

A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion

Beauregard

Padgett

Parshad

2017

Journal of Computational and Applied Mathematics

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“…In our approach, we prefer to replace the unknown y ℓ+1 in the source function by a suitable approximation, such as that obtained by using an Euler or extrapolation method [1,7,16].…”

Section: Eikonal Transformation Based Exponential Splittingmentioning

confidence: 99%

“…. , (1.6) where w α , g α are mesh functions for approximating v(z α ), g(v α , z α ), respectively [1,8]. Further, while a typical parallel exponential splitting is…”

Section: Introductionmentioning

confidence: 98%

“…. , are constructed through an adaptive mechanism [1,8]. Then the solution of (1.2) can be expressed as v(z ℓ+1 ) = exp…”

Section: Introductionmentioning

confidence: 99%

“…Splitting strategies have been backbones of many highly straightforward yet accurate numerical methods for solving linear or nonlinear partial differential equations, in particular those oriented from important multiphysics applications or multiscale environments [1][2][3][4]. In a splitting process, the underlying modeling equation is decomposed into several more durable subproblems to solve before a final approximation is obtained.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers

Sheng

Sun

2014

Computers & Mathematics with Applications

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a b s t r a c tThis paper explores applications of the exponential splitting method for approximating highly oscillatory solutions of the n-dimensional paraxial Helmholtz equation. An eikonal transformation is introduced for oscillation-free platforms and matrix operator decompositions. It is found that the sequential, parallel and combined exponential splitting formulas possess not only anticipated algorithmic simplicity and efficiency, but also the accuracy and asymptotic stability required for highly oscillatory wave computations.

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A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains

Beauregard

Sheng

2013

Numerical Methods Partial

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This article studies a fully adaptive finite difference method for solving quenching‐type nonlinear reaction‐diffusion equations over circular domains. Although an auxiliary condition at the origin and radial symmetry are imposed, adaptations are accomplished via arc‐length‐based monitoring functions in space and time, respectively. The monotonicity and positivity of the numerical solution are proved following a suitable grid constraint, and the numerical stability is ensured in the von Neumann sense. Theoretical bounds of the critical quenching radius are obtained and then refined through the computation. Computational examples are provided to illustrate the effectiveness and plausibility of the new adaptive computational procedure developed. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 472–489, 2014

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An adaptive splitting approach for the quenching solution of reaction–diffusion equations over nonuniform grids

Cited by 26 publications

References 19 publications

A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion

A nonlinear splitting algorithm for systems of partial differential equations with self-diffusion

Exponential splitting for n-dimensional paraxial Helmholtz equation with high wavenumbers

A fully adaptive approximation for quenching‐type reaction‐diffusion equations over circular domains

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