4 research outputs found
Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids
Finite volume schemes often have difficulties to resolve the low Mach number
(incompressible) limit of the Euler equations. Incompressibility is only
non-trivial in multiple spatial dimensions. Low Mach fixes, however generally
are applied to the one-dimensional method and the method is then used in a
dimensionally split way. This often reduces its stability. Here, it is
suggested to keep the one-dimensional method as it is, and only to extend it to
multiple dimensions in a particular, all-speed way. This strategy is found to
lead to much more stable numerical methods. Apart from the conceptually
pleasing property of modifying the scheme only when it becomes necessary, the
multi-dimensional all-speed extension also does not include any free parameters
or arbitrary functions, which generally are difficult to choose, or might be
problem dependent. The strategy is exemplified on a Lagrange Projection method
and on a relaxation solver
Truly multi-dimensional all-speed schemes for the Euler equations on Cartesian grids
Finite volume schemes often have difficulties to resolve the low Mach number (incompressible) limit of the Euler equations. Incompressibility is only non-trivial in multiple spatial dimensions. Low Mach fixes, however generally are applied to the one-dimensional method and the method is then used in a dimensionally split way. This often reduces its stability. Here, it is suggested to keep the one-dimensional method as it is, and only to extend it to multiple dimensions in a particular, all-speed way. This strategy is found to lead to much more stable numerical methods. Apart from the conceptually pleasing property of modifying the scheme only when it becomes necessary, the multi-dimensional all-speed extension also does not include any free parameters or arbitrary functions, which generally are difficult to choose, or might be problem dependent. The strategy is exemplified on a Lagrange Projection method and on a relaxation solver
Exact solution and the multidimensional Godunov scheme for the acoustic equations
The acoustic equations derived as a linearization of the Euler equations are
a valuable system for studies of multi-dimensional solutions. Additionally they
possess a low Mach number limit analogous to that of the Euler equations.
Aiming at understanding the behaviour of the multi-dimensional Godunov scheme
in this limit, first the exact solution of the corresponding Cauchy problem in
three spatial dimensions is derived. The appearance of logarithmic
singularities in the exact solution of the 4-quadrant Riemann Problem in two
dimensions is discussed. The solution formulae are then used to obtain the
multidimensional Godunov finite volume scheme in two dimensions. It is shown to
be superior to the dimensionally split upwind/Roe scheme concerning its domain
of stability and ability to resolve multi-dimensional Riemann problems. It is
shown experimentally and theoretically that despite taking into account
multi-dimensional information it is, however, not able to resolve the low Mach
number limit