A Space-Time Flux Reconstruction Approach for Unsteady 3D Parabolic Equations with Enhanced Convergence Acceleration

The space-time flux reconstruction (FR) approach featuring hyperbolic reformulation of second-order terms is extended to three spatial dimensions and its performance with locally
adaptive pseudo-time stepping schemes is assessed. High-order accurate schemes are generated via the tensor product of the Gauss-Legendre points, for use on 4D hypercube elements. Eigendecomposition of the reformulated equations demonstrates the hyperbolic properties of the new system, in contrast to the parabolic character of the original governing equation. While conventional parabolic schemes can be severely restricted by O(h2) time steps necessary for numerical stability, it is shown that the new approach remains stable for O(h) pseudo-time steps. It was verified that the target order-of-accuracy (OOA) was achieved for schemes involving three spatial dimensions and that O(h) pseudo-time steps were realised. Solving unsteady flow problems in a steady context, with respect to pseudo-time, also allows for the implementation of locally adaptive pseudo-time stepping (LAPTS). This can radically accelerate convergence to the pseudo-steady state for the linear problems investigated compared to traditional time marching procedures, increasing the competitiveness of the new approach for practical CFD analysis.