Greedy Trial Subspace Selection in Meshfree Time-Stepping Scheme with Applications in Coupled Bulk-Surface Pattern Formations

TL;DR

This paper introduces a greedy trial subspace selection method for meshfree time-stepping schemes that effectively balances spatial and temporal errors, demonstrated through coupled bulk-surface pattern formation simulations.

Contribution

It proposes a novel greedy algorithm for selecting trial centers in kernel-based collocation methods, improving efficiency and accuracy in solving parabolic PDEs.

Findings

Reduces trial space dimensions significantly

Maintains high accuracy in coupled bulk-surface simulations

Balances spatial and temporal discretization errors effectively

Abstract

Combining kernel-based collocation methods with time-stepping methods to solve parabolic partial differential equations can potentially introduce challenges in balancing temporal and spatial discretization errors. Typically, using kernels with high orders of smoothness on some sufficiently dense set of trial centers provides high spatial approximation accuracy that can exceed the accuracy of finite difference methods in time. The paper proposes a greedy approach for selecting trial subspaces in the kernel-based collocation method applied to time-stepping to balance errors in both well-conditioned and ill-conditioned scenarios. The approach involves selecting trial centers using a fast block-greedy algorithm with new stopping criteria that aim to balance temporal and spatial errors. Numerical simulations of coupled bulk-surface pattern formations, a system involving two functions in the domain and two on the boundary, illustrate the effectiveness of the proposed method in reducing trial space dimensions while maintaining accuracy.