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This paper reports a recent development of the high-order spectral difference method with divergence cleaning (SDDC) for accurate simulations of both ideal and resistive magnetohydrodynamics (MHD) on curved unstructured grids consisting of high-order isoparametric quadrilateral elements. The divergence cleaning approach is based on the improved generalized Lagrange multiplier, which is thermodynamically consistent. The SDDC method can achieve an arbitrarily high order of accuracy in spatial discretization, as demonstrated in the test problems with smooth solutions. The high-order SDDC method combined with the artificial dissipation method can sharply capture shock interfaces with the oscillation-free property and resolve small-scale vortex structures and density fluctuations on relatively sparse grids. The robustness of the codes is demonstrated through long time simulations of ideal MHD problems with progressively interacting shock structures, resistive MHD problems with high Lundquist numbers, and viscous resistive MHD problems on complex curved domains.

A 3D parallel high-order spectral difference (SD) solver with curved local mesh refinement is developed in this research to simulate flow through stenoses of varied degrees (50%, 60%, 65%, 70% and 75%) of radius constriction at inlet Reynolds number of 500. This solver employs high-order curved mesh in the vicinity of arterial wall and the local mesh refinement technique reduces the overall computational cost by distributing more elements in critical regions. In simulation of flow through stenosis of 50% radius constriction, velocity profiles predicted from the SD solver agree well with previous DNS results and experimental data. Mesh independency study shows that numerical results from a conforming and a non-conforming mesh agree well with each other. When the constriction degree is larger than 50%, visualizations through iso-surfaces of Q-criterion show that vortex rings are ejected from the stenosis throat, advecting downstream before they hit the vessel walls and they finally break down and merge into a large bulk region of small-scale turbulence. The observation is consistent with the vorticity contour which is characterized by development of the Kelvin-Helmholtz instability when shear layers are formed, rolled up and advected downstream between the central jet and the recirculation region. When the constriction degree turns to 75%, the flow transitions rapidly downstream of stenosis throat and dramatic pressure drop is witnessed. This provides a fluid-dynamic explanation for clinical definition of critical stenosis (i.e. over 75% luminal radius narrowing). Furthermore, pressure drop across a stenosis is found to be proportional to square of ratio of non-stenosed area to minimum area at the stenosis throat with a linear correlation coefficient equal to 0.9998. Finally, this solver is proven to have excellent scalability on massively parallel computers when multi-level refinement of meshes is performed to capture small-scale structures in the turbulence region.

When using the discontinuous Galerkin or Spectral Difference (SD) method to discretize ideal magnetohydrodynamic (MHD) equations, it is challenging to satisfy the divergence-free constraint for the magnetic field over long-period time integration. To tackle this challenge, the SD method is integrated with an unstaggered Constrained Transport approach (SDCT). In addition to solving the two-dimensional ideal MHD equations, one more equation describing the transport of the magnetic potential is introduced. After each time step, the magnetic field will be updated by computing the curl of the magnetic potential. This strategy preserves ∇ · B = 0 exactly by construction in the discrete sense. Meanwhile, the additional computational cost is only approximately 20% more than that without the constrained transport. Moreover, the inclusion of the constrained transport does not obstruct the implementation of the artificial viscosity for shock capturing. Several well-known benchmark test cases are studied in this paper using the SDCT method. In the magnetic field loop advection test, the proposed SDCT avoids spurious growth of magnetic energy, and the numerical dissipation is shown to decrease when increasing the polynomial degree while maintaining the total degrees of freedom. In the propagation of Alfvén wave problem, the high-order accuracies of the SDCT method are verified. In the Orzag-Tang vortex problem, the predicted pressure distribution and density contours match well with those in the reference [1]. Meanwhile, a mesh convergence study shows that the SDCT method equipped with the artificial viscosity terms can produce converged results even in the vicinity of shocks.

We present two major improvements over the Compressible High-ORder Unstructured Spectral difference (CHORUS) code published in Wang et al., “A compressible high-order unstructured spectral difference code for stratified convection in rotating spherical shells,” J. Comput. Phys. 290, 90–111 (2015). The new code is named CHORUS++ in this paper. Subsequently, we perform a series of efficient simulations for rotationally constrained convection (RCC) in spherical shells. The first improvement lies in the integration of the high-order spectral difference method with a boundary-conforming transfinite mapping on cubed-sphere grids, thus ensuring exact geometric representations of spherical surfaces on arbitrary sparse grids. The second improvement is on the adoption of higher-order elements (sixth-order) in CHORUS++ vs third-order elements for the original CHORUS code. CHORUS++ enables high-fidelity RCC simulations using sixth-order elements on very coarse grids. To test the accuracy and efficiency of using elements of different orders, CHORUS++ is applied to a laminar solar benchmark, which is characterized by columnar banana-shaped convective cells. By fixing the total number of solution degrees of freedom, the computational cost per time step remains unchanged. Nevertheless, using higher-order elements in CHORUS++ resolves components of the radial energy flux much better than using third-order elements. To obtain converged predictions, using sixth-order elements is 8.7 times faster than using third-order elements. This significant speedup allows global-scale fully compressible RCC simulations to reach equilibration of the energy fluxes on a small cluster of just 40 cores. In contrast, CHORUS simulations were performed by Wang et al. on supercomputers using approximately 10 000 cores. Using sixth-order elements in CHORUS++, we further carry out global-scale solar convection simulations with decreased rotational velocities. Interconnected networks of downflow lanes emerge and surround broader and weaker regions of upflow fields. A strong inward kinetic energy flux compensated by an enhanced outward enthalpy flux appears. These observations are all consistent with those published in the literature. Furthermore, CHORUS++ can be extended to magnetohydrodynamic simulations with potential applications to the hydromagnetic dynamo processes in the interiors of stars and planets.