[Paper Review] A universal solution scheme for fractional and classical PDEs

Yixuan Wu, Yanzhi Zhang|arXiv (Cornell University)|Jan 30, 2021

Numerical methods in engineering42 references4 citations

TL;DR

This paper proposes a unified meshless pseudospectral method based on generalized inverse multiquadric (GIMQ) radial basis functions to solve classical and fractional PDEs with the operator $(-\Delta)^{\frac{\alpha}{2}}$ for $\alpha \in (0,2]$. By analytically expressing the Laplacian of GIMQ functions via the Gauss hypergeometric function, the method unifies discretization of local and nonlocal Laplacians, avoids numerical approximation of hypersingular integrals, and enables high accuracy with fewer unknowns and reduced computational cost across dimensions.

ABSTRACT

We propose a unified meshless method to solve classical and fractional PDE problems with $(-Δ)^{\fracα{2}}$ for $α\in (0, 2]$. The classical ($α= 2$) and fractional ($α< 2$) Laplacians, one local and the other nonlocal, have distinct properties. Therefore, their numerical methods and computer implementations are usually incompatible. We notice that for any $α\ge 0$, the Laplacian $(-Δ)^{\fracα{2}}$ of generalized inverse multiquadric (GIMQ) functions can be analytically written by the Gauss hypergeometric function, and thus propose a GIMQ-based method. Our method unifies the discretization of classical and fractional Laplacians and also bypasses numerical approximation to the hypersingular integral of fractional Laplacian. These two merits distinguish our method from other existing methods for the fractional Laplacian. Extensive numerical experiments are carried out to test the performance of our method. Compared to other methods, our method can achieve high accuracy with fewer number of unknowns, which effectively reduces the storage and computational requirements in simulations of fractional PDEs. Moreover, the meshfree nature makes it free of geometric constraints and enables simple implementation for any dimension $d \ge 1$. Additionally, two approaches of selecting shape parameters, including condition number-indicated method and random-perturbed method, are studied to avoid the ill-conditioning issues when large number of points.

Motivation & Objective

• To develop a unified numerical framework that can solve both classical ($\alpha=2$) and fractional ($\alpha<2$) PDEs with the same computational scheme.
• To overcome the incompatibility between existing numerical methods for local and nonlocal Laplacians, which require separate implementations.
• To eliminate the need for numerical approximation of the hypersingular integral in fractional Laplacian computation.
• To enable efficient, high-accuracy simulations of fractional PDEs with reduced storage and computational costs.

Proposed method

• The method employs generalized inverse multiquadric (GIMQ) radial basis functions with power $\beta = -(d+1)/2$ for dimension $d \geq 1$.
• It leverages the analytical expression of $(-\Delta)^{\frac{\alpha}{2}}$ applied to GIMQ functions via the Gauss hypergeometric function, enabling exact computation without numerical quadrature.
• The method uses a pseudospectral collocation approach, where the solution is approximated as a linear combination of GIMQ basis functions centered at scattered nodes.
• Dirichlet boundary conditions are enforced directly on the domain boundary ($\partial\Omega$) for $\alpha=2$ or on the complement ($\Omega^c$) for $\alpha<2$, using the pointwise definition of the fractional Laplacian.
• Two shape parameter selection strategies are introduced: condition-number-indicated and random-perturbed methods, to mitigate ill-conditioning with large point sets.
• The approach is meshfree, allowing straightforward implementation in any dimension $d \geq 1$ without geometric constraints.

Experimental results

Research questions

• RQ1 Can a single numerical scheme be developed to unify the solution of classical and fractional PDEs governed by $(-\Delta)^{\frac{\alpha}{2}}$?
• RQ2 How can the nonlocal nature of the fractional Laplacian be accurately captured without resorting to numerical integration of hypersingular integrals?
• RQ3 What is the impact of boundary conditions on the solution domain when the boundary is not in contact with the domain, particularly in fractional diffusion?
• RQ4 How do different shape parameter selection strategies affect the conditioning and accuracy of the GIMQ-based method for large-scale problems?
• RQ5 To what extent does the proposed method outperform existing RBF or Gaussian-based methods in terms of accuracy and computational efficiency?

Key findings

• The proposed method achieves spectral accuracy comparable to the recent Gaussian-based method, but with fewer unknowns and reduced computational cost.
• The GIMQ-based method avoids numerical evaluation of the hypersingular integral in the fractional Laplacian by using analytical expressions via the Gauss hypergeometric function.
• For uniformly distributed centers, the optimal shape parameter in the Gaussian-based method is more sensitive to the number of points than in the GIMQ-based method.
• The condition-number-indicated shape parameter selection effectively suppresses ill-conditioning and maintains accuracy even with large numbers of points, though it requires additional computational time.
• The random-perturbed shape parameter method reduces computation time while still maintaining acceptable accuracy and stability.
• Numerical results show that fractional Laplacian solutions are significantly influenced by boundary conditions even when the boundary region is disconnected from the domain, due to the nonlocal nature of the operator, with stronger effects observed for smaller $\alpha$ or shorter distances.