Algorithms and Data Structures for Multi-Adaptive Time-Stepping

TL;DR

This paper introduces algorithms and data structures for multi-adaptive time-stepping in differential equations, enabling efficient, component-wise adaptive time steps with demonstrated performance improvements on benchmark problems.

Contribution

It presents novel algorithms and data structures for efficient multi-adaptive time-stepping, extending Galerkin methods for differential equations.

Findings

Efficient algorithms for recursive time slab construction.

Adaptive time step selection improves computational efficiency.

Demonstrated performance on benchmark problems.

Abstract

Multi-adaptive Galerkin methods are extensions of the standard continuous and discontinuous Galerkin methods for the numerical solution of initial value problems for ordinary or partial differential equations. In particular, the multi-adaptive methods allow individual and adaptive time steps to be used for different components or in different regions of space. We present algorithms for efficient multi-adaptive time-stepping, including the recursive construction of time slabs and adaptive time step selection. We also present data structures for efficient storage and interpolation of the multi-adaptive solution. The efficiency of the proposed algorithms and data structures is demonstrated for a series of benchmark problems.