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Conservative polynomial approximations and applications to Fokker-Planck equations

Tino Laidin, Lorenzo Pareschi

TL;DR

The paper develops a constrained $L^2$-minimization framework to construct moment-preserving, spectrally accurate approximations based on orthogonal polynomials. It derives explicit formulas for a conservative projection that preserves chosen moments and proves spectral convergence under standard smoothness assumptions. The authors apply the method to several Fokker-Planck and related kinetic problems on bounded and unbounded domains, demonstrating exact moment conservation and improved long-time behavior compared to unconstrained schemes. The approach provides a versatile, structure-preserving tool for high-accuracy simulations in kinetic theory and beyond, with potential extensions to Vlasov-type equations and socio-economic models.

Abstract

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.

Conservative polynomial approximations and applications to Fokker-Planck equations

TL;DR

The paper develops a constrained -minimization framework to construct moment-preserving, spectrally accurate approximations based on orthogonal polynomials. It derives explicit formulas for a conservative projection that preserves chosen moments and proves spectral convergence under standard smoothness assumptions. The authors apply the method to several Fokker-Planck and related kinetic problems on bounded and unbounded domains, demonstrating exact moment conservation and improved long-time behavior compared to unconstrained schemes. The approach provides a versatile, structure-preserving tool for high-accuracy simulations in kinetic theory and beyond, with potential extensions to Vlasov-type equations and socio-economic models.

Abstract

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.
Paper Structure (22 sections, 3 theorems, 92 equations, 17 figures, 7 tables)

This paper contains 22 sections, 3 theorems, 92 equations, 17 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Conservative approximations by orthogonal polynomials
  3. Orthogonal polynomials
  4. Approximation of moments
  5. Unbounded domains
  6. Moment preserving orthogonal polynomials
  7. Numerical examples and applications
  8. Test 1: Approximation of functions
  9. Bounded domain: $[-1,1]$
  10. Unbounded domain: $\mathbb{R}$
  11. Unbounded domain: $\mathbb{R}^+$
  12. Test 2: Kinetic Fokker-Planck equation
  13. Spectral accuracy and conservations
  14. Long-time behaviour
  15. Test 3: A model of Opinion formation
  16. ...and 7 more sections

Key Result

Theorem 2.1

\newlabelthm:ClassicalFourierSpectralAccuracy0 If $f\in H_\omega^r(\Omega)$, where $r \geq 0$ is an integer, then there exists a constant $C > 0$ dependent on $\alpha,\beta$ and $r$ such that

Figures (17)

  • Figure 1: Spectral radius of the matrix $M^{-1}$ as a function of $N$ (Left), and as a function of $Q$ (Right) for Chebyshev 2nd kind polynomials.
  • Figure 1: Constrained Legendre approximation for \ref{['testfunctionBounded']}.
  • Figure 2: Constrained approximations for Chebyshev 1st kind (Top Left), Chebyshev 2nd kind (Top Right) and Jacobi $\alpha=1$, $\beta=-1/2$ (Bottom) for \ref{['testfunctionBounded']}.
  • Figure 3: Condition number of the matrix M (Left), and minimum of the condition number of the matrices $\hat{C}_k$ (Right) as functions of $Q$ for Chebyshev 2nd kind polynomials
  • Figure 4: Constrained Hermite approximation for \ref{['testfunctionHermite']}.
  • ...and 12 more figures

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 2.2
  • Definition 2.3
  • Theorem 2.5
  • Proof 1
  • Remark 2.6
  • Remark 2.7