Roe‐type Riemann solvers for general hyperbolic systems

Abstract

SUMMARYWe present a Roe‐type weak formulation Riemann solver where the average coefficient matrix is computed numerically. The novelty of this approach is that it is general enough that can be applied to any hyperbolic system while retaining the accuracy of the original Roe solver. We show applications to the compressible Euler equations with general equation of state. An alternative version of the method uses directly the eigenvectors in the averaging process, simplifying the algorithm. These new solvers are applied in conservative and path‐conservative schemes with high‐order accuracy and on unstructured meshes. Copyright © 2014 John Wiley & Sons, Ltd.

Polynomial viscosity methods for multispecies kinematic flow models

“…The first alternative, that yields characteristic-wise numerical schemes, gives better resolution than the second, since the amount of numerical viscosity is much smaller. The prototype method of this kind of is Roe's method [1] (see also [2]), for which the first-order numerical flux, specified here for two adjacent states L and R to the left and right of a cell boundary, is given bŷ…”

“…The first alternative, that yields characteristic‐wise numerical schemes, gives better resolution than the second, since the amount of numerical viscosity is much smaller. The prototype method of this kind of is Roe's method (see also ), for which the first‐order numerical flux, specified here for two adjacent states and to the left and right of a cell boundary, is given by where is a Roe matrix (related to the flux Jacobian ) for the flux and denotes a real diagonalizable matrix that is computed through the eigenvectors and eigenvalues of (not to be confused with the matrix of absolute values of the entries of ). Conversely, the second alternative, which gives rise to component‐wise schemes, tends to yield faster methods.…”

Roe‐type Riemann solver for gas–liquid flows using drift‐flux model with an approximate form of the Jacobian matrix

“…Integrating the matrices d U / d Z and d F / d Z in ξ , for each state φ ( ξ , u L , u R ) reaches φ ( z L + ξ ( z R − z L )) to obtain and Ĉ , see . Except the first and second lines of d U / d Z matrix and the first line of d F / d Z , all other terms were integrated numerically using a Gauss–Legendre quadrature with three points, employing a procedure similar to one found in .…”

“…The schemes provide the intercell fluxes by a family of waves which propagates through the cell's interface. For two‐phase flows the resultant non‐linear hyperbolic system of the conservative laws is frequently solved employing Roe linearization , see for example .…”

“…The main advantage of the analytical Jacobian matrix is related to the fact that it does not have to be recalculated for each cell at each time step, thereby reducing the computational cost. Because of the algebraic complexity of the Roe‐type matrix, some of its elements were constructed with a Gauss Legendre quadrature with three points, as presented in .…”