Convergence of Implicit Difference Scheme for 1D Lagrangian Hydrodynamics coupled to Radiation Transport Equation

TL;DR

This paper presents a fully implicit finite difference scheme for 1D hydrodynamics coupled with radiation transport, demonstrating improved accuracy and convergence over semi-implicit methods in radiation hydrodynamics simulations.

Contribution

The paper introduces a fully implicit scheme for radiation hydrodynamics that improves accuracy and convergence, with detailed benchmarking and comparison to semi-implicit methods.

Findings

The implicit scheme agrees well with known scaling laws for shock propagation.

It reduces errors significantly compared to semi-implicit methods.

The method's convergence improves with smaller time steps, at a modest computational cost.

Abstract

A fully implicit finite difference scheme has been developed to solve the hydrodynamic equations coupled with radiation transport. Solution of the time dependent radiation transport equation is obtained using the discrete ordinates method and the energy flow into the Lagrangian meshes as a result of radiation interaction is fully accounted for. A tridiagonal matrix system is solved at each time step to determine the hydrodynamic variables implicitly. The results obtained from this fully implicit radiation hydrodynamics code in the planar geometry agrees well with the scaling law for radiation driven strong shock propagation in aluminium. For the point explosion problem the self similar solutions are compared with results for pure hydrodynamic case in spherical geometry and the effect of radiation energy transfer is determined. Having, thus, benchmarked the code, convergence of the method w.r.t. time step is studied in detail and compared with the results of commonly used semi-implicit method. It is shown that significant error reduction is feasible in the implicit method in comparison to the semi-implicit method, though at the cost of slightly more CPU time.