High-Order CENO Finite-Volume Scheme with Anisotropic Adaptive Mesh Refinement: Efficient Inexact Newton Method for Steady Three-Dimensional Flows

Abstract

A high-order finite-volume scheme with anisotropic adaptive mesh refinement (AMR) is combined with a parallel inexact Newton method for the solution of steady compressible fluid flows governed by the Euler and Navier–Stokes equations on three-dimensional multi-block body-fitted hexahedral meshes. The proposed steady flow solution method combines a family of robust and accurate high-order central essentially non-oscillatory (CENO) spatial discretization schemes with both a scalable and efficient Newton–Krylov–Schwarz (NKS) algorithm and a block-based anisotropic AMR method. The CENO scheme is based on a hybrid solution reconstruction procedure that provides high-order accuracy in smooth regions (even for smooth extrema) and non-oscillatory transitions at discontinuities and makes use of a high-order representation of the mesh and a high-order treatment of boundary conditions. In the proposed Newton method, the resulting linear systems of equations are solved using the generalized minimal residual (GMRES) algorithm preconditioned by a domain-based additive Schwarz technique. The latter uses the domain decomposition provided by the block-based AMR scheme leading to a fully parallel implicit approach with an efficient scalability of the overall scheme. The anisotropic AMR method is based on a binary tree data structure and permits local anisotropic refinement of the grid in preferred directions as directed by appropriately specified physics-based refinement criteria. Numerical results are presented for a range of inviscid and viscous steady problems and the computational performance of the combined scheme is demonstrated and assessed.

Appendix A. Appendix

1.1 A.1 Trilinear Transformation

The trilinear transformation is defined by

(A.1)

where p, q and r are the Cartesian coordinates in the canonical space of the reference cube. The transformation vector coefficients for the trilinear transformation are defined by ${\mathbf{T_1}}$, ${\mathbf{T_2}}$, ${\mathbf{T_3}}$, ${\mathbf{T_4}}$, ${\mathbf{T_5}}$, ${\mathbf{T_6}}$, ${\mathbf{T_7}}$ and ${\mathbf{T_8}}$ which are in turn given by

$$\begin{aligned} \begin{array}{ll} \textbf{T}_1 &{} = 1/8 \ ({\mathbf{N_1}} + {\mathbf{N_2}} + {\mathbf{N_3}} + {\mathbf{N_4}} + {\mathbf{N_5}} + {\mathbf{N_6}} + {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_2 &{} = 1/8 \ ({\mathbf{N_2}} - {\mathbf{N_1}} - {\mathbf{N_3}} - {\mathbf{N_4}} + {\mathbf{N_5}} + {\mathbf{N_6}} - {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_3 &{} = 1/8 \ ({\mathbf{N_3}} - {\mathbf{N_2}} - {\mathbf{N_1}} - {\mathbf{N_4}} + {\mathbf{N_5}} - {\mathbf{N_6}} + {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_4 &{} = 1/8 \ ({\mathbf{N_4}} - {\mathbf{N_2}} - {\mathbf{N_3}} - {\mathbf{N_1}} - {\mathbf{N_5}} + {\mathbf{N_6}} + {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_5 &{} = 1/8 \ ({\mathbf{N_1}} - {\mathbf{N_2}} - {\mathbf{N_3}} + {\mathbf{N_4}} + {\mathbf{N_5}} - {\mathbf{N_6}} - {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_6 &{} = 1/8 \ ({\mathbf{N_1}} - {\mathbf{N_2}} + {\mathbf{N_3}} - {\mathbf{N_4}} - {\mathbf{N_5}} + {\mathbf{N_6}} - {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_7 &{} = 1/8 \ ({\mathbf{N_1}} + {\mathbf{N_2}} - {\mathbf{N_3}} - {\mathbf{N_4}} - {\mathbf{N_5}} - {\mathbf{N_6}} + {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \textbf{T}_8 &{} = 1/8 \ ({\mathbf{N_2}} - {\mathbf{N_1}} + {\mathbf{N_3}} + {\mathbf{N_4}} - {\mathbf{N_5}} - {\mathbf{N_6}} - {\mathbf{N_7}} + {\mathbf{N_8}}),\\ \end{array} \end{aligned}$$

where ${\mathbf{N_1}}$, ${\mathbf{N_2}}$, ${\mathbf{N_3}}$, ${\mathbf{N_4}}$, ${\mathbf{N_5}}$, ${\mathbf{N_6}}$, ${\mathbf{N_7}}$ and ${\mathbf{N_8}}$ are the vertices of the hexahedron as depicted in Fig. 24a. The tangent vectors to the coordinate lines are defined by

Fig. 24

a Second-order accurate trilinear representation of generic curved geometry. The black nodes at the vertices represent the second-order accurate basis functions. b Fourth-order accurate tricubic representation of generic curved geometry. The blue nodes denote the extra nodes (Color figure online)

1.2 A.2 Tricubic Transformation

The tricubic transformation is defined by

(A.2)

and the corresponding tangent vectors to the coordinate lines are defined by

The transformation vector coefficients is this case are given by

Here ${\mathbf{N_1}}$–${\mathbf{N_8}}$ are the vertices of the hexahedron and ${\mathbf{N_9}}$–${\mathbf{N_32}}$ are the high-order nodes as depicted in Fig. 24b.

1.3 A.3 Transformation Jacobians

The determinants of the Jacobians for volume and surface integration are given by

(A.3)