A Non-iterative Overlapping Schwarz Waveform Relaxation Algorithm for Wave Equation

TL;DR

This paper introduces a non-iterative Schwarz waveform relaxation algorithm for wave equations, leveraging wave velocity limits to achieve rapid, accurate solutions with only three steps per time span, promising improved scalability and speed.

Contribution

The paper presents the first non-iterative overlapping Schwarz waveform relaxation algorithm for wave equations, based on physical wave velocity constraints, reducing computational steps and enhancing efficiency.

Findings

RSWR achieves high accuracy with only 3 steps per time span.

Numerical experiments confirm RSWR's potential for scalability and fast convergence.

The algorithm is grounded in the physical principle of limited wave velocity, enabling non-iterative computation.

Abstract

The Schwarz Waveform Relaxation algorithm (SWR) exchanges the waveform of boundary value between neighbouring sub-domains, which provides a more efficient way than the other Schwarz algorithms to realize distributed computation. However, the convergence speed of the traditional SWR is slow, and various optimization strategies have been brought in to accelerate the convergence. In this paper, we propose a non-iterative overlapping variant of SWR for wave equation, which is named Relative Schwarz Waveform Relaxation algorithm (RSWR). RSWR is inspired by the physical observation that the velocity of wave is limited, based on the Theory of Relativity. The change of value at one space point will take time span $\Delta t$ to transmit to another space point and vice versa. This $\Delta t$ could be utilized to design distributed numerical algorithm, as we have done in RSWR. During each time span, RSWR needs only 3 steps to achieve high accurate waveform, by using the predict-select-update strategy. The key for this strategy is to find the maximum time span for the waveform. The validation of RSWR could be proved straightfowardly. Numerical experiments show that RSWR is accurate, and is potential to be scalable and fast.