保守多项式逼近及其在福克-普朗克方程中的应用
Conservative polynomial approximations and applications to Fokker-Planck equations
AI总结：
本文提出一种保留函数任意矩且不失谱收敛性的保守正交多项式逼近方法，推导结构保持的偏微分方程数值方法并通过福克-普朗克方程应用验证其通用性与性能。
摘要：
我们研究基于正交多项式构造逼近的问题，该逼近需保留给定函数的任意矩集合且不丢失谱收敛性。为此，我们针对正交多项式的通用基计算约束最优逼近多项式。此构造完全通用，能推导结构保持的偏微分方程数值方法，这些方法需守恒解的某些矩（通常代表问题的相关物理量）。这些性质对高精度捕捉解的长期行为至关重要。我们通过多个福克-普朗克方程的数值应用，展示了该方法的通用性与性能。
英文摘要：
We address the problem of constructing approximations based on orthogonal polynomials that preserve an arbitrary set of moments of a given function without loosing the spectral convergence property. To this aim, we compute the constrained polynomial of best approximation for a generic basis of orthogonal polynomials. The construction is entirely general and allows us to derive structure preserving numerical methods for partial differential equations that require the conservation of some moments of the solution, typically representing relevant physical quantities of the problem. These properties are essential to capture with high accuracy the long-time behavior of the solution. We illustrate with the aid of several numerical applications to Fokker-Planck equations the generality and the performances of the present approach.
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