An implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L2 norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.

中文翻译：

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Fisher-KPP 方程的结构保持不连续 Galerkin 格式

建议并分析了具有无通量边界条件的人口密度的 Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) 方程的隐式 Euler 不连续 Galerkin 格式。使用指数变量变换，数值方案自动保持离散解的正性。推导出离散熵不等式，如果初始熵小于域的测度，则证明离散密度在L1范数下对稳定稳态的指数时间衰减。当网格大小趋于零时，离散解被证明在 L2 范数中收敛到时间离散 Fisher-KPP 方程的唯一强解。在一维空间中的数值实验说明了理论结果。