# Maximum-principle preserving space-time isogeometric analysis

## TL;DR

This paper introduces a nonlinear stabilization method for space-time isogeometric analysis that preserves the maximum principle, allows high-order discretizations without CFL constraints, and reduces computational costs through a partitioned scheme.

## Contribution

It presents a novel graph-theoretic artificial diffusion stabilization technique with a shock detector, unconditionally preserving the discrete maximum principle for arbitrary order.

## Key findings

- The method ensures solutions satisfy the discrete maximum principle unconditionally.
- A partitioned space-time scheme reduces computational costs while maintaining stability.
- A twice differentiable stabilization scheme improves nonlinear convergence.

## Abstract

In this work we propose a nonlinear stabilization technique for convection-diffusion-reaction and pure transport problems discretized with space-time isogeometric analysis. The stabilization is based on a graph-theoretic artificial diffusion operator and a novel shock detector for isogeometric analysis. Stabilization in time and space directions are performed similarly, which allow us to use high-order discretizations in time without any CFL-like condition. The method is proven to yield solutions that satisfy the discrete maximum principle (DMP) unconditionally for arbitrary order. In addition, the stabilization is linearity preserving in a space-time sense. Moreover, the scheme is proven to be Lipschitz continuous ensuring that the nonlinear problem is well-posed. Solving large problems using a space-time discretization can become highly costly. Therefore, we also propose a partitioned space-time scheme that allows us to select the length of every time slab, and solve sequentially for every subdomain. As a result, the computational cost is reduced while the stability and convergence properties of the scheme remain unaltered. In addition, we propose a twice differentiable version of the stabilization scheme, which enjoys the same stability properties while the nonlinear convergence is significantly improved. Finally, the proposed schemes are assessed with numerical experiments. In particular, we considered steady and transient pure convection and convection-diffusion problems in one and two dimensions.