Analysis of Semi-Implicit Time Integration Schemes For Direct Numerical Simulation of Turbulent Convection in Liquid Metals

Fully explicit time integration schemes are very inefficient for numerical simulation of diffusion dominated problems. In case of natural convection flow in liquid metals an implicit treatment of the thermal diffusion terms allows for the use of substantially increased time steps without involving loss of physically relevant information. Two suitable semi-implicit time integration schemes are investigated analytically by a Von Neumann stability analysis and a spectral analysis of the numerical error. Numerical solutions by the semi-implicit schemes are compared to the exact solution of a 1D linear test problem. The results show the crucial influence of the discretization ratio X = At/Ax on the accuracy of the numerical solutions. First 3D time dependent numerical simulations of natural convection in liquid metals with the semi-implicit time integration schemes confirm the theoretically estimated gain in the time step width and result in CPU-time savings up to a factor of 50 compared to the fully explicit scheme.