The effect of artificial viscosity on numerical boundary feedback control of linear hyperbolic systems

TL;DR

This paper investigates how artificial viscosity affects the numerical boundary feedback control of linear hyperbolic systems, revealing that numerical diffusion influences decay rates and stability in discretized schemes.

Contribution

It provides a detailed analysis of artificial viscosity effects on upwind schemes for hyperbolic systems, including decay rate behavior depending on CFL conditions and solution derivatives.

Findings

Decay rates depend on CFL number and second derivatives of data.

Upwind scheme with CFL=1 achieves expected decay rates.

Numerical results confirm analytical decay behavior.

Abstract

A numerical analysis of the effect of artificial viscosity is undertaken in order to understand the effect of numerical diffusion on numerical boundary feedback control. The analysis is undertaken on the linear hyperbolic systems discretised using the upwind scheme. The upwind scheme solves the advection-diffusion equation with up to second-order accuracy. The analysis shows that the upwind scheme with CFL equal to one gives the expected theoretical decay up to first-order. On the other hand the upwind scheme with CFL less than one gives decay depending on the second derivative of the data and the CFL number. Further the decay rates deteriorate if the second derivatives of the solution are small. Thus the decay rates computed by the numerical schemes tend to be higher in comparison to the theoretical prediction. Computations on test cases which include isothermal Euler and the St Venant Equations confirm the analytical results.