dorsal/arxiv
View SchemaValidation and Calibration of Models for Reaction-Diffusion Systems
| Authors | Rui Dilao, Joaquim Sainhas |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9712007 |
| URL | https://arxiv.org/abs/patt-sol/9712007 |
| DOI | 10.1142/S0218127498000917 |
Abstract
Space and time scales are not independent in diffusion. In fact, numerical simulations show that different patterns are obtained when space and time steps ($\Delta x$ and $\Delta t$) are varied independently. On the other hand, anisotropy effects due to the symmetries of the discretization lattice prevent the quantitative calibration of models. We introduce a new class of explicit difference methods for numerical integration of diffusion and reaction-diffusion equations, where the dependence on space and time scales occurs naturally. Numerical solutions approach the exact solution of the continuous diffusion equation for finite $\Delta x$ and $\Delta t$, if the parameter $\gamma_N=D \Delta t/(\Delta x)^2$ assumes a fixed constant value, where $N$ is an odd positive integer parametrizing the alghorithm. The error between the solutions of the discrete and the continuous equations goes to zero as $(\Delta x)^{2(N+2)}$ and the values of $\gamma_N$ are dimension independent. With these new integration methods, anisotropy effects resulting from the finite differences are minimized, defining a standard for validation and calibration of numerical solutions of diffusion and reaction-diffusion equations. Comparison between numerical and analytical solutions of reaction-diffusion equations give global discretization errors of the order of $10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum relative random deviation from the spherical symmetry of the order of 0.2%, and the standard deviation of the fluctuations around the mean circular wave front is of the order of $10^{-3}$.
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"abstract": "Space and time scales are not independent in diffusion. In fact, numerical\nsimulations show that different patterns are obtained when space and time steps\n($\\Delta x$ and $\\Delta t$) are varied independently. On the other hand,\nanisotropy effects due to the symmetries of the discretization lattice prevent\nthe quantitative calibration of models. We introduce a new class of explicit\ndifference methods for numerical integration of diffusion and\nreaction-diffusion equations, where the dependence on space and time scales\noccurs naturally. Numerical solutions approach the exact solution of the\ncontinuous diffusion equation for finite $\\Delta x$ and $\\Delta t$, if the\nparameter $\\gamma_N=D \\Delta t/(\\Delta x)^2$ assumes a fixed constant value,\nwhere $N$ is an odd positive integer parametrizing the alghorithm. The error\nbetween the solutions of the discrete and the continuous equations goes to zero\nas $(\\Delta x)^{2(N+2)}$ and the values of $\\gamma_N$ are dimension\nindependent. With these new integration methods, anisotropy effects resulting\nfrom the finite differences are minimized, defining a standard for validation\nand calibration of numerical solutions of diffusion and reaction-diffusion\nequations. Comparison between numerical and analytical solutions of\nreaction-diffusion equations give global discretization errors of the order of\n$10^{-6}$ in the sup norm. Circular patterns of travelling waves have a maximum\nrelative random deviation from the spherical symmetry of the order of 0.2%, and\nthe standard deviation of the fluctuations around the mean circular wave front\nis of the order of $10^{-3}$.",
"arxiv_id": "patt-sol/9712007",
"authors": [
"Rui Dilao",
"Joaquim Sainhas"
],
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],
"doi": "10.1142/S0218127498000917",
"title": "Validation and Calibration of Models for Reaction-Diffusion Systems",
"url": "https://arxiv.org/abs/patt-sol/9712007"
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