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| This talk deals with nonlinear diffusion problems. The framework is so general as to include the Stefan problem, porous medium equation and several kinds of cross-diffusion systems. We show that the solutions of the nonlinear diffusion problems can be approximated by those of semilinear reaction-diffusion systems. This indicates that the mechanism of nonlinear diffusion might be captured by reaction-diffusion interaction. The reaction-diffusion systems include only simple reactions and linear diffusions. Resolving semilinear problems is typically easier than dealing with nonlinear problems. Therefore, our ideas are expected to reveal effective approaches to the study of nonlinear problems. Applying the similinear approixmation to numerical analysis, we constructed and analyzed a linear numerical scheme for the nonlinear diffusion systems. The linear algorithm is a very easy to implement scheme. We derive optimal rates of convergence of the linear scheme by means of the reaction-diffusion system approximation. |
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