A Fourth-Order Solution-Adaptive CENO Scheme for Space-Physics Flows on Three-Dimensional Multi-Block Cubed-Sphere Grids

A high-order central essentially non-oscillatory (CENO) finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of hyperbolic conservation laws on three-dimensional cubed-sphere grids. In particular, the fluid flows of interest are governed by the compressible form of Euler and ideal magnetohydrodynamics (MHD) equations and pertain to space-physics applications. The adaptive cubedsphere simulation framework represents a flexible design based on a genuine multi-block implementation, leading to high-order accuracy, flux calculation, adaptivity and parallelism that are consistent throughout the domain, including at the boundaries between the sectors of the cubed-sphere grid. 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. Furthermore, the CENO formulation is naturally uniformly high-order on the whole cubed-sphere grid including at sector boundaries and corners. The scheme is applied in combination with a divergence correction technique to enforce the solenoidal condition for the magnetic field. A fourth-order quadrature rule is derived and employed in transferring the solution content between levels of coarseand fine-grained mesh resolution so as to preserve the high-order character of the solution while minimizing the computational effort and interprocessor communication. Several numerical results are presented and discussed to demonstrate the accuracy and current capabilities of the three-dimensional, high-order, solution-adaptive CENO computational framework for multi-block cubed-sphere grids. 1 SCOPE OF CURRENT WORK High-order accurate and efficient computational methods are highly desirable in many fields of computational physics, especially in the study of problems characterized by a wide range of temporal and length scales on which the interesting physics occurs, such as the global magnetohydrodynamics (MHD) modelling of space-physics problems. To obtain accurate and more affordable solutions of such complex flows, the usage of both high-order discretizations and adaptive mesh refinement (AMR) is often required. Adaptive mesh refinement is an effective approach for coping with the computational cost of large-scale numerical simulations, such as those encountered in space physics. For geometries involving spherical objects, which is usually the case in space physics flows, second-order schemes in combination with various AMR strategies have been successfully developed and widely used on Cartesian and latitude-longitude spherical (or spherical-polar) grids to enhance local solution features [13–15, 20, 21, 29, 33, 36–38]. Moreover, AMR strategies for discretizations of spherical domains based on cubed-sphere grids [31], which recently emerged as an attractive alternative to sphericalpolar grids, have also been proposed in both twodimensions (2D) by St-Cyr et al. [32] and, later on, in three-dimensions (3D) by Ivan et al. [18]. An important characteristic of the latter approach is that rather than mapping the cubed-sphere grid to a Cartesian computational domain (a typical approach for 2D spheres [8, 32]), the employed strategy is based on a multi-block approach with unstructured root connectivity, previously proposed by Gao and Groth [12], thereby providing a multi-block AMR finite-volume framework that is more general and not restricted Figure 1: Complex cut into an adapted cubed-sphere grid showing block boundaries and associated meshes. only to cubed-sphere topologies. An example of an adapted, isotropically-refined, cubed-sphere grid that has been obtained with the AMR framework described in [18] is shown in Fig. 1. Very recently, Williamschen and Groth [41] developed a second-order anisotropic refinement strategy for multi-block hexahedral grids and applied it to cubed-sphere grids as well. Despite significant advances in new discretization schemes of various types for 3D hyperbolic conservation laws, including finite difference methods (e.g., [26]), discontinuous Galerkin methods (e.g., [6, 24, 40]), finite-volume methods (e.g. [1, 2, 8, 22]), and combinations of these approaches (e.g., [10]), most advanced MHD frameworks for parallel space-physics simulations are only second-order accurate [21,36,38]. Recently, Ivan et al. [16, 19] have extended the high-order central essentially non-oscillatory (CENO) finite-volume schemes [17, 34] to 3D geometries with structured hexahedral grids, with particular emphasis on cubed-sphere mesh topologies. Although the formulation is general and suitable for solution accuracies of different orders, the emphasis was on providing fourth-order accuracy. The scope of our current work is to further advance the computational framework by combining the 3D CENO approach [16, 19] with the block-based AMR strategy described in [18], and provide a first assessment on the predictive capabilities of the proposed 3D high-order AMR framework for solutions of hyperbolic conservation laws. The overarching objective of the research is to develop a flexible, scalable and efficient 3D high-order framework that can be applied in combination with (an)isotropic mesh refinement to various space-physics flows. A first technical challenge in the application of CENO scheme to cubed-sphere grids is the high-order treatment of hexahedral cells with non-planar cell faces, that are present in cubed-sphere discretizations. A second major technical challenge that must be addressed is how to deal with reconstruction stencils at degenerate block edges and corners. Near these regions the numerical scheme must adapt to deviations from the regular mesh topology and find efficiently the right neighbours to be included in the supporting reconstruction stencil. A third challenge is the formulation of efficient algorithms to transfer solution content accurately between blocks of different grid resolution. Consistent algorithms have been developed to address all these challenges efficiently, thereby allowing the application of the CENO scheme to multi-block cubed-sphere grids and the carrying out of large-scale simulations on massively-parallel computing clusters with distributed memory architecture. The remainder of the paper provides a brief description of the proposed high-order solution-adaptive CENO algorithm along with a selected summary of numerical results. 2 FOURTH-ORDER CENO METHOD ON 3D CUBED-SPHERE GRIDS Herein, nonlinear conservation laws of the form ∂tU+~∇ ·~F = S+Q (1) are considered, where U is the vector of conserved variables, ~F is the flux dyad, and S and Q are numerical and physical source terms that may arise for certain equation sets and application problems. In particular, this paper discusses the development and application of the fourth-order adaptive CENO method on cubedsphere grids for the cases of the MHD and Euler equations. For MHD using the generalized Lagrange multiplier (GLM) approach to control divergence errors, as in [9,34], U is given by U = [ ρ, ρ~V , ~B, ρe, ψ ]T , where ρ is the gas density, ~V = (u,v,w) is the velocity, ~B = (Bx,By,Bz) is the magnetic field, ρe is the total energy and ψ is the generalized Lagrange multiplier employed to control errors in the divergence of the magnetic field. Here, the total energy is given by ρe= p/(γ−1)+ρV 2/2+B2/2, where V and B are the magnitudes of the velocity and magnetic field vectors, respectively, and γ is the ratio of specific heats. The flux dyad,~F, is given by