High-Order Spline Upwind for Space-Time Isogeometric Analysis

TL;DR

This paper introduces a novel isogeometric space-time method for the heat equation that employs high-order splines and a stabilization technique to ensure stability and accuracy in numerical solutions.

Contribution

The paper presents a new space-time isogeometric approach with a stabilization term based on high-order artificial diffusions, improving stability and maintaining optimal accuracy.

Findings

Method demonstrates enhanced stability in numerical experiments

Achieves high accuracy with stabilized high-order splines

Linear system becomes lower block-triangular, simplifying computations

Abstract

We propose an innovative isogeometric space-time method for the heat equation, with smooth splines approximation in both space and time. To enhance the stability of the method we add a stabilizing term, based on a linear combination of high-order artificial diffusions. This term is designed in order to make the linear system lower block-triangular, that is, lower triangular with respect to time. In order to keep optimal accuracy, the stabilization terms are further weighted in terms of the residual. Through a series of numerical experiments, we validate the method's capability, showcasing its stability and accuracy.