Volumetric radial basis function methods applied to gas dynamics

A set of rotational and translation transformations are applied to the Euler gas dynamic equations. In such a transformed coordinate frame, the partial differential equations (PDEs) appear as a set of steady ordinary differential equations (ODEs) in the rotating, translating frame. By using appropriate linear combinations of the ODEs, we obtain a transformed set of ODEs that resemble the compatibility equations from the method of characteristics plus additional terms for the angular momentum or streamline bending. The new dependent variables are cast into radial basis functions that are volumetrically integrated over each piecewise continuous subregion. At discontinuities such as shocks or contact surfaces, these discontinuities are propagated by the Rankine-Hugoniot jump conditions. For the case of weak shocks that are not important to track, they are captured and dampened away by the use of artificial viscosity. Knots over each continuous subregion may be added, deleted, or redistributed while constraining the appropriate volumetric dependent variables to be strictly conservative. Because volumetric integration is a smoothing operation, the numerical solutions converge faster compared with simple collocation.