High-order methods for parabolic equations using a space-time flux reconstruction approach

A novel implementation of the flux reconstruction (FR) approach featuring a hyperbolic reformulation of the governing equations in space-time is presented for viscous linear and non-linear flow problems in both one and two spatial dimensions. The procedure generates high-order accurate schemes — in both space and time — that can analyse diffusion-type equations by recasting second-order equations as first-order systems. Conventional high-order accurate analysis of parabolic equations is severely restricted by limits on time step, which can be avoided by reformulation into a system of hyperbolic equations, with the caveat that only steady solutions may be considered. However, the computation of the resulting system within the space-time FR framework permits the high-order accuracy analysis of unsteady flows with rapid convergence to the steady state, in the pseudo-time sense, within a procedure that can be implemented in a straightforward manner for diffusion-type problems. Eigendecomposition is used to demonstrate that the new systems are hyperbolic in nature for both the 1D and 2D Advection-Diffusion Equations. The development and successful implementation of first-order space-time FR schemes for the 1D and 2D Diffusion Equations is illustrated. It is also verified that the target order-of-accuracy (OOA) is achieved for schemes involving both one and two spatial dimensions. An application of the space-time flux reconstruction approach to the Euler Equations is presented and discussed, with a view to future implementation to the Unsteady Navier-Stokes Equations with similar hyperbolic reformulation of viscous terms.