Spline-based solution transfer for space-time methods in 2D+t

TL;DR

This paper presents a novel HCT spline-based solution transfer method for 2D+t space-time finite element simulations, achieving high accuracy, mass conservation, and smooth visualization capabilities.

Contribution

It introduces a new transfer process using HCT splines that ensures high-order accuracy, maximum principle preservation, and smooth surrogate solutions for space-time methods.

Findings

Achieves up to 4th order accuracy.

Maintains a discrete maximum principle.

Provides a smooth, continuous surrogate solution.

Abstract

This work introduces a new solution-transfer process for slab-based space-time finite element methods. The new transfer process is based on Hsieh-Clough-Tocher (HCT) splines and satisfies the following requirements: (i) it maintains high-order accuracy up to 4th order, (ii) it preserves a discrete maximum principle, (iii) it asymptotically enforces mass conservation, and (iv) it constructs a smooth, continuous surrogate solution in between space-time slabs. While many existing transfer methods meet the first three requirements, the fourth requirement is crucial for enabling visualization and boundary condition enforcement for space-time applications. In this paper, we derive an error bound for our HCT spline-based transfer process. Additionally, we conduct numerical experiments quantifying the conservative nature and order of accuracy of the transfer process. Lastly, we present a qualitative evaluation of the visualization properties of the smooth surrogate solution.