Partitioning methods for reaction–diffusion problems

Heineken, Wolfram; Warnecke, Gerald

doi:10.1016/j.apnum.2005.09.001

Cited by 10 publications

(15 citation statements)

References 25 publications

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“…the transformed Riemann problem reads (18) is also a shock solution of (4)- (6). The type of the shock is the same as in (17).…”

Section: Theorem 43 For Everyûmentioning

confidence: 96%

“…For example, for the choice 0 = · · · = lev max −1 = 0 one will obtain a uniform grid of level lev max . A similar algorithm to steer the structure of the grid was proposed in [6], the only difference being that there a scaled Z 2 indicator was used instead of the gradient indicator proposed here.…”

Section: Grid Adaptionmentioning

confidence: 97%

“…With the initial conditions given in Example 2 we define (5), (6). The error will be measured at any time t n , n = 1, .…”

Section: Definition Of Errorsmentioning

confidence: 99%

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The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

Heineken

Kunik

2008

Journal of Computational and Applied Mathematics

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A finite volume method with grid adaption is applied to two hyperbolic problems: the ultra-relativistic Euler equations, and a scalar conservation law. Both problems are considered in two space dimensions and share the common feature of moving shock waves. In contrast to the classical Euler equations, the derivation of appropriate initial conditions for the ultra-relativistic Euler equations is a non-trivial problem that is solved using one-dimensional shock conditions and the Lorentz invariance of the system. The discretization of both problems is based on a finite volume method of second order in both space and time on a triangular grid. We introduce a variant of the min-mod limiter that avoids unphysical states for the Euler system. The grid is adapted during the integration process. The frequency of grid adaption is controlled automatically in order to guarantee a fine resolution of the moving shock fronts. We introduce the concept of "width refinement" which enlarges the width of strongly refined regions around the shock fronts; the optimal width is found by a numerical study. As a result we are able to improve efficiency by decreasing the number of adaption steps. The performance of the finite volume scheme is compared with several lower order methods.

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“…the transformed Riemann problem reads (18) is also a shock solution of (4)- (6). The type of the shock is the same as in (17).…”

Section: Theorem 43 For Everyûmentioning

confidence: 96%

Section: Grid Adaptionmentioning

confidence: 97%

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The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

Heineken

Kunik

2008

Journal of Computational and Applied Mathematics

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“…and heat transfer has led to physically interesting and mathematically challenging problems related to the nontrivial numerical solution of the singularly perturbed initial/boundary value problem [1][2][3][4][5][6][7][8][9]. In general, solution of the singularly perturbed problem contains high-gradient boundary layers along the boundary of the domain [2].…”

mentioning

confidence: 99%

“…During the past decade, many techniques of the finite element method for solving the singularly perturbed problems have been proposed and investigated [2,4,7,8,11,12]. The conventional Galerkin finite element method usually fails to analyze such unsteady reaction-diffusion problems when either the diffusion coefficient becomes small or a small time step is employed in the time discretization.…”

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confidence: 99%

Finite volume element method for analysis of unsteady reaction–diffusion problems

Phongthanapanich

Dechaumphai

2009

Acta Mech Sin

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A finite volume element method is developed for analyzing unsteady scalar reaction-diffusion problems in two dimensions. The method combines the concepts that are employed in the finite volume and the finite element method together. The finite volume method is used to discretize the unsteady reaction-diffusion equation, while the finite element method is applied to estimate the gradient quantities at cell faces. Robustness and efficiency of the combined method have been evaluated on uniform rectangular grids by using available numerical solutions of the two-dimensional reaction-diffusion problems. The numerical solutions demonstrate that the combined method is stable and can provide accurate solution without spurious oscillation along the highgradient boundary layers.

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W‐schemes in spectral methods for reaction‐diffusion equations

Pulch

2024

Proc Appl Math and Mech

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We consider reaction‐diffusion equations, which are parabolic systems of partial differential equations. Assuming periodic boundary conditions in space, a multidimensional Fourier transform can be used. A spectral method yields an initial value problem of a stiff system of ordinary differential equations for time‐dependent Fourier coefficients. We investigate the application of Rosenbrock‐Wanner W‐schemes in the time integration of the ordinary differential equations including step size selection. Results of numerical computations are presented using a Turing system in two space dimensions.

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Partitioning methods for reaction–diffusion problems

Cited by 10 publications

References 25 publications

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

The analytical solution of two interesting hyperbolic problems as a test case for a finite volume method with a new grid refinement technique

Finite volume element method for analysis of unsteady reaction–diffusion problems

W‐schemes in spectral methods for reaction‐diffusion equations

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