This work contains the derivation and type analysis of the conical ideal magnetohydrodynamic equations. The 3D ideal MHD equations with Powell source terms, subject to the assumption that the solution is conically invariant, are projected onto a unit sphere using tools from tensor calculus. Conical flows provide valuable insight into supersonic and hypersonic flow past bodies, but are simpler to analyze and solve numerically. Previously, work has been done on conical inviscid flows governed by the Euler equations with great success. It is known that some flight regimes involve flows of ionized gases, and thus there is motivation to extend the study of conical flows to the case where the gas is electrically conducting. To the authors’ knowledge, the conical magnetohydrodynamic equations have never been derived, and so this paper is the first investigation of that system. Among the results, we show that conical flows for this case do exist mathematically and that the governing system of partial differential equations is of mixed type. Throughout the domain it can be either hyperbolic or elliptic depending on the solution.

In this work, we systematically derive the governing equations of supersonic conical flow by projecting the 3D Euler equations onto the unit sphere. These equations result from taking the assumption of conical invariance on the 3D flow field. Under this assumption, the compressible Euler equations reduce to a system defined on the surface of the unit sphere. This compressible flow problem has been successfully used to study the steady supersonic flow past cones of arbitrary cross section by reducing the number of spatial dimensions from three down to two while still capturing many of the relevant 3D effects. In this paper, the powerful machinery of tensor calculus is utilized to avoid reference to any particular coordinate system. With the flexibility to use any coordinate system on the surface of a sphere, the equations can be more readily solved numerically when a structured mesh is used by defining the mesh lines to be the coordinate lines. The type of the system of partial differential equations would be hyperbolic or elliptic based on whether the crossflow Mach number is supersonic or subsonic.

The Boltzmann equation is essential to the accurate modeling of rarefied gases. Unfortunately, traditional numerical solvers for this equation are too computationally expensive for many practical applications. With modern interest in hypersonic flight and plasma flows, to which the Boltzmann equation is relevant, there would be immediate value in an efficient simulation method. The collision integral component of the equation is the main contributor of the large complexity. A plethora of new mathematical and numerical approaches have been proposed in an effort to reduce the computational cost of solving the Boltzmann collision integral, yet it still remains prohibitively expensive for large problems. This paper aims to accelerate the computation of this integral via machine learning methods. In particular, we build a deep convolutional neural network to encode/decode the solution vector, and enforce conservation laws during post-processing of the collision integral before each time-step. Our preliminary results for the spatially homogeneous Boltzmann equation show a drastic reduction of computational cost. Specifically, our algorithm requires O(n3) operations, while asymptotically converging direct discretization algorithms require O(n6), where n is the number of discrete velocity points in one velocity dimension. Our method demonstrated a speed up of 270 times compared to these methods while still maintaining reasonable accuracy.

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