论文标题
非线性交叉扩散系统的订购模型减少
Reduced order modelling of nonlinear cross-diffusion systems
论文作者
论文摘要
在这项工作中，我们为人口动力学的非线性交叉扩散问题提供了一个减少阶数模型，用于shigesada-kawasaki-teramoto（SKT）方程，并带有Lotka-volterra动力学。 SKT方程在空间中的有限差分离散化导致线性 - 季度的普通微分方程（ODE）系统。还原订单模型（ROM）具有与完整阶模型（FOM）相同的线性二次结构。使用ROM的线性二次结构，将减少订单的溶液与适当的正交分解（POD）独立于完整溶液中计算。通过施加张力舱，进一步加速了还原溶液的计算。 SKT方程的图案的形成由快速瞬态相和一个较长的稳态相组成。减少的订单解决方案是通过将时间分为两个时间间隔来计算的。在数值实验中，我们显示了具有模式形成的一维SKT方程，即以时间窗口形式获得的减少阶溶液，即主体分解框架（P-POD）比在整个时间间隔中获得的全球POD溶液（G-POD）更准确。此外，我们通过简化的溶液在数值上显示了熵的减小，这对于非线性交叉扩散方程（例如SKT方程）的整体存在很重要。
In this work, we present a reduced-order model for a nonlinear cross-diffusion problem from population dynamics, for the Shigesada-Kawasaki-Teramoto (SKT) equation with Lotka-Volterra kinetics. The finite-difference discretization of the SKT equation in space results in a system of linear--quadratic ordinary differential equations (ODEs). The reduced order model (ROM) has the same linear-quadratic structure as the full order model (FOM). Using the linear-quadratic structure of the ROM, the reduced-order solutions are computed independent of the full solutions with the proper orthogonal decomposition (POD). The computation of the reduced solutions is further accelerated by applying tensorial POD. The formation of the patterns of the SKT equation consists of a fast transient phase and a long steady-state phase. Reduced order solutions are computed by separating the time, into two-time intervals. In numerical experiments, we show for one-and two-dimensional SKT equations with pattern formation, the reduced-order solutions obtained in the time-windowed form, i.e., principal decomposition framework (P-POD), are more accurate than the global POD solutions (G-POD) obtained in the whole time interval. Furthermore, we show the decrease of the entropy numerically by the reduced solutions, which is important for the global existence of nonlinear cross-diffusion equations such as the SKT equation.
