Implicit Quadrature-Free Direct Reconstruction Method for Efficient Scale-Resolving Simulations (Communications in Computational Physics, 2023)
The present study develops implicit physical domain-based discontinuous Galerkin (DG) methods for efficient scale-resolving simulations on mixed-curvedmeshes. Implicit methods are essential to handle stiff systems in many scale-resolvingsimulations of interests in computational science and engineering. The physicaldomain-based DG method can achieve high-order accuracy using the optimal bases setand preserve the required accuracy on non-affine meshes. When using the quadrature-based DG method, these advantages are overshadowed by severe computational costson mixed-curved meshes, making implicit scale-resolving simulations unaffordable.To address this issue, the quadrature-free direct reconstruction method (DRM) is extended to the implicit DG method. In this approach, the generalized reconstructionapproximates non-linear flux functions directly in the physical domain, making thecomputing-intensive numerical integrations precomputable at a preprocessing step.The DRM operator is applied to the residual computation in the matrix-free method.The DRM operator can be further extended to the system matrix computation for thematrix-explicit Krylov subspace method and preconditioning. Finally, the A-stableRosenbrock-type Runge–Kutta methods are adopted to achieve high-order accuracyin time. Extensive verification and validation from the manufactured solution to implicit large eddy simulations are conducted. The computed results confirm that theproposed method significantly improves computational efficiency compared to thequadrature-based method while accurately resolving detailed unsteady flow featuresthat are hardly captured by scale-modeled simulations.
Keywords: Direct reconstruction method; implicit Runge–Kutta method; quadrature-free method; discontinuous Galerkin method; implicit large eddy simulation; high performance computing.