Finite-volume Treatment of Dispersion-relation- Preserving and Optimized Prefactored Compact Schemes for Wave Propagation

In developing suitable numerical techniques for computational aero-acoustics, the DispersionRelation-Preserving (DRP) scheme by Tam and coworkers, and the optimized prefactored compact (OPC) scheme by Ashcroft and Zhang have shown desirable properties of reducing both dissipative and dispersive errors. These schemes, originally based on the finite difference, attempt to optimize the coefficients for better resolution of short waves with respect to the computational grid while maintaining pre-determined formal orders of accuracy. In the present study, finite volume formulations of both schemes are presented to better handle the nonlinearity and complex geometry encountered in many engineering applications. Linear and nonlinear wave equations, with and without viscous dissipation, have been adopted as the test problems. Highlighting the principal characteristics of the schemes and utilizing linear and nonlinear wave equations with different wavelengths as the test cases, the performance of these approaches is documented. For the linear wave equation, there is no major difference between the DRP and OPC schemes. For the nonlinear wave equations, the finite volume version of both DRP and OPC schemes offer substantially better solutions in regions of high gradient or discontinuity. Nomenclature c sound speed (Eq.(1), (7), (8)) CFL Courant – Friedrichs – Lewy number = Di derivative in point i (Eq.(35)) Di, Di Forward and Backward derivative operators in the point i (Eqs.(24) (26), (33), (34)) E error (Eqs.(4), (27), (78) (80)) fd finite difference fv finite volume K function computed in the stage i, in the Runge-Kutta time integration (Eqs.(46),(47)) Pe Peclet number = si, eN-i coefficients used to compute the derivative operator on the boundary, for 3 OPC scheme (Eq.(33), (34), (37), (38)) t time ui, ui forward, and backward operator computed on east face in the cell i (Eqs. (29)-(32), (57)(60), (66) (69)) ui, ui forward, and backward operator computed on west face in the cell i (Eqs. (29)-(32), (57)(60), (66) (69)) ui the value of parameter u on face e, in cell i (Eq.(35), (36), (55), (56)(89), (91)) ui the value of parameter u on face w, in cell i (Eq.(35), (36), (55), (56)(89), (91)) u the value of function u in the stage m in the Runge-Kutta time integration (Eqs. (46),(47)) u the value of function u in the n iteration, it is related to time integration (Eqs. (41)(42), (46) (47)) x, y coordinate in space Δx, Δy, Δt length of grid in space, in x and y direction, respecyively time step size , wavenumber of a space marching scheme (Eqs.(3), (4), (27) ) α wavenumber (Eq.(3), (70)) μ viscosity (Eq.(92)-(96)) ω angular frequency (Eqs. (42), (43),(72), (73)) (), () parameter designed for forward, respectively backward operator (Eqs.(25), (26), (37), (38))