AN IMPLICIT BLOCK JACOBI APPROACH FOR HIGH-ORDER FLUX RECONSTRUCTION METHOD

Recently, the flux reconstruction (FR) method has attracted more and more attentions for its simplicity and generality. However, it is still computationally expensive and time consuming when simulating the complex flow problems by FR method. There is a huge demand for developing appropriate efficient implicit solvers and parallel computing techniques for FR. This paper proposes an implicit high-order flux reconstruction solver on GPU platform based on the block Jacobi iteration method. As it is inefficient to solve the large global linear system resulting from spatial and implicit temporal discretization of FR directly. A block Jacobi approach is used to change the characteristics of the lift-hand matrix of the global linear system and this avoids the dependence of neighboring elements. Therefore, only the diagonal blocks of global matrix need to be stored and calculated. Then, the problem of solving the huge global linear system is transformed into solving a series of local linear equations simultaneously. Finally, these small local linear equations would be solved by the LU decomposition method in parallel on GPU platforms. Two typical cases, including subsonic flows over a bump and a NACA0012 airfoil, were simulated and compared with the multi-grid explicit Runge-Kutta scheme. The numerical results demonstrated that the present implicit method can reduce the iterations significantly. Meanwhile, the implicit solver has shown at least 10x speedup over the multi-grid Runge-Kutta scheme in all cases.