[Paper Review] Numerical investigations of traveling singular sources problems via moving mesh method

Zhicheng Hu, Keiwei Liang|arXiv (Cornell University)|Mar 8, 2012

Fluid Dynamics and Vibration Analysis22 references3 citations

TL;DR

This paper proposes a moving mesh method combined with domain decomposition to solve one-dimensional PDEs with multiple traveling singular sources moving at different speeds. By splitting the domain into subdomains around each source and solving local mesh equations independently, the method avoids computing jumps in the solution derivative, enabling second-order spatial convergence and efficient parallel computation. The approach successfully captures blow-up phenomena for sources with varying velocities, a case not previously numerically studied.

ABSTRACT

This paper studies the numerical solution of traveling singular sources problems. In such problems, a big challenge is the sources move with different speeds, which are described by some ordinary differential equations. A predictor-corrector algorithm is presented to simulate the position of singular sources. Then a moving mesh method in conjunction with domain decomposition is derived for the underlying PDE. According to the positions of the sources, the whole domain is splitted into several subdomains, where moving mesh equations are solved respectively. On the resulting mesh, the computation of jump $[\dot{u}]$ is avoided and the discretization of the underlying PDE is reduced into only two cases. In addition, the new method has a desired second-order of the spatial convergence. Numerical examples are presented to illustrate the convergence rates and the efficiency of the method. Blow-up phenomenon is also investigated for various motions of the sources.

Motivation & Objective

• To address the numerical challenge of solving PDEs with time-dependent singular sources moving at different speeds.
• To overcome the failure of standard finite difference/element methods when crossing moving source positions due to solution derivative jumps.
• To develop a robust, second-order convergent numerical scheme that avoids explicit computation of jump terms [u_x] at source locations.
• To enable efficient parallel computation through domain decomposition and local mesh equidistribution on subdomains.
• To investigate the blow-up behavior of solutions under various source motion patterns, including non-uniform and periodic velocities.

Proposed method

• A predictor-corrector algorithm is used to track the time-dependent positions α_i(t) of q singular sources governed by ODEs dα_i/dt = ψ_i(t, α_i(t), u).
• The computational domain is decomposed into q+1 subdomains separated by the q sources, with local moving mesh equations solved independently on each subdomain.
• The moving mesh is generated via MMPDEs (e.g., MMPDE4/5/6) based on the equidistribution principle to concentrate mesh points near sources.
• The physical PDE is discretized using a finite difference scheme on the resulting adaptive mesh, with only two distinct stencils needed due to the domain decomposition.
• The method avoids direct computation of [u_x] jumps by treating each subdomain independently, simplifying the discretization process.
• A special time-stepping implementation ensures second-order spatial convergence of the overall scheme.

Experimental results

Research questions

• RQ1 Can a moving mesh method with domain decomposition effectively handle PDEs with multiple singular sources moving at different, non-uniform speeds?
• RQ2 How can the jump in the solution derivative [u_x] at moving sources be avoided in the numerical discretization?
• RQ3 What is the convergence rate of the proposed method in space, and can second-order accuracy be preserved?
• RQ4 Does the method accurately capture the blow-up phenomenon when sources move at different velocities?
• RQ5 Can the method be extended to multiple sources (q > 2) with different motion patterns?

Key findings

• The proposed method achieves second-order spatial convergence under a specific time-stepping strategy, as confirmed by numerical experiments.
• For a single source moving at speed k=2, blow-up occurs at t ≈ 2.0397 at position x ≈ 4.0794 with u_max ≈ 3.16×10^6.
• For two sources moving with the same speed k=2, blow-up occurs at t ≈ 1.6896 on the second source with u_max ≈ 3.16×10^6.
• In the symmetric periodic motion case with A=π, blow-up occurs simultaneously on both sources at t ≈ 2.4969 with u_max ≈ 3.16×10^6.
• The method successfully simulates blow-up for sources with different velocities, a case not previously numerically investigated.
• Using local absorbing boundary conditions allows for a smaller observed domain without affecting solution accuracy, as demonstrated in numerical results.