Stabilized Isogeometric Collocation Methods for Hyperbolic Conservation Laws

TL;DR

This paper presents stabilized isogeometric collocation methods for hyperbolic conservation laws, combining residual-based viscosity and projection stabilization to handle shocks and maintain high accuracy with reduced computational cost.

Contribution

The authors develop a novel stabilized collocation scheme that effectively manages shocks and oscillations while enabling efficient high-order discretizations.

Findings

Robust handling of shocks in hyperbolic laws

High-order accuracy preserved on smooth problems

Reduced computational cost for high-order schemes

Abstract

We introduce stabilized spline collocation schemes for the numerical solution of nonlinear, hyperbolic conservation laws. A nonlinear, residual-based viscosity stabilization is combined with a projection stabilization-inspired linear operator to stabilize the scheme in the presence of shocks and prevent the propagation of spurious, small-scale oscillations. Due to the nature of collocation schemes, these methods possess the possibility for greatly reduced computational cost of high-order discretizations. Numerical results for the linear advection, Burgers, Buckley-Leverett, and Euler equations show that the scheme is robust in the presence of shocks while maintaining high-order accuracy on smooth problems.