Eigenvalues and eigenvectors of the Euler equations in general geometries

Abstract

The complete eigensystem, including eigenvalues and left and right eigenvectors, of the Euler equations of inviscid flow are derived in a general finite volume coordinate frame. The symmetry of the eigenvector space is demonstrated from a mathematical and geometric viewpoint. Results are presented for 2-D and 3-D inviscid flow, and their application in computational fluid dynamics (CFD) is discussed.

Very-high-order weno schemes

“…The L 1 -norm of the error [4] e L1 reaches almost immediately the theoretical rate-of-convergence [2,4] r cnvrg L1 ¼ 2r À 1, as the number of cells N c ¼ N x À 1 ¼ N y À 1 increases. The result is significant in that it provides systematic verification of previous observation by Balsara and Shu [3] that the unsplit 2-D linewise extension of the veryhigh-order upwind and WENOM schemes returns the theoretical order-of-accuracy of the 1-D tests, for the scalar linear advection equation (48).…”

“…The Godunov flux is computed using an exact Riemann solver, where the tangential-to-the-cell-interface velocity is treated as a passively convected quantity [43, pp. 149-150] of the left and right eigenvectors used to define the local characteristic variables (Section 5.3) are given, for the general 3-D case with arbitrary cell-interface orientation, in [48] (they can be simplified in an ifless construction for the 2-D case [12,48], but we used the general 3-D expressions in the present work). The resulting semi-discrete scheme is integrated in time using the SSPRK(3, 3) method of Shu and Osher [23].…”

“…The performance and order-of-accuracy of the various schemes is studied by comparing with the exact solution of the 2-D advection equation (48) for different initial conditions (ICs). All of the computations used the corresponding 'SSPRKð2r; 2r À 1Þ time-discretization [37][38][39].…”

Implicit finite element schemes for the stationary compressible Euler equations

“…Applying the chain rule, one transforms the Euler equations to with the Jacobian tensor Both Jacobians ∂ F ( x ) /∂ U and ∂ F ( y ) /∂ U are diagonalizable with a complete set of eigenvectors, which makes the Euler equations hyperbolic. The eigenvectors, eigenvalues, and Jacobians can be found in 36.…”

Assessment of WENO schemes for multi‐dimensional Euler equations using GPU

“…The eigenvalues represent the wave speeds in each direction, where is the sound speed. The matrices of left and right eigenvectors ( L and R = L −1 ) can be found in . To handle both the negative and positive wave speeds, we use both component‐wise and characteristic‐wise Lax–Friedrichs flux splitting .…”