Local-Maximum-and-Minimum-Preserving Solution Remapping Technique to Accelerate Flow Convergence for Discontinuous Galerkin Methods in Shape Optimization Design

In this work, a solution remapping technique is developed to accelerate the flow convergence for the intermediate shapes when a high-order discontinuous Galerkin (DG) method is employed as a compressible Euler flow solver in the airfoil design problems. Once the shape is updated, the proposed technique is applied to initialize the flow simulation for the new shape via a solution remapping formula and a maximum-and-minimum-preserving limiter. First, the solution remapping formula is used to remap the solution of the current shape into a piecewise polynomial on the mesh of the new shape. Then the piecewise polynomial is constrained with the maximum-and-minimum-preserving limiter. The modified piecewise polynomial is used as the initial value for the new shape. Numerical experiments show that the proposed technique can attractively accelerate flow convergence and significantly reduce up to 80% of the computational time in the airfoil design problems with a high-order DG solver.