Discrete adjoint computations for relaxation Runge-Kutta methods

TL;DR

This paper derives the discrete adjoint of relaxation Runge-Kutta methods for entropy-stable discretizations of nonlinear conservation laws, ensuring properties like time-symmetry and aiding optimal control applications.

Contribution

It introduces the discrete adjoint formulation for relaxation Runge-Kutta schemes and proves time-symmetry preservation for linear skew-symmetric systems.

Findings

Discrete adjoint preserves entropy dissipation properties.

Time-symmetry is maintained in linear skew-symmetric systems.

Proper treatment of the relaxation parameter is crucial for accurate adjoint computations.

Abstract

Relaxation Runge-Kutta methods reproduce a fully discrete dissipation (or conservation) of entropy for entropy stable semi-discretizations of nonlinear conservation laws. In this paper, we derive the discrete adjoint of relaxation Runge-Kutta schemes, which are applicable to discretize-then-optimize approaches for optimal control problems. Furthermore, we prove that the derived discrete relaxation Runge-Kutta adjoint preserves time-symmetry when applied to linear skew-symmetric systems of ODEs. Numerical experiments verify these theoretical results while demonstrating the importance of appropriately treating the relaxation parameter when computing the discrete adjoint.