Compact Central Scheme Research Articles

A class of high-order weighted compact central schemes for solving hyperbolic conservation laws

We propose a class of weighted compact central schemes for solving hyperbolic conservation laws. The linear version can be considered as a high-order extension of the central Lax–Friedrichs scheme and the central conservation element and solution element scheme. On every cell, the solution is approximated by a Pth-order polynomial of which all the DOFs are stored and updated separately. The cell average is updated by a classical finite volume scheme which is constructed based on space-time staggered meshes such that the fluxes are continuous across the interfaces of the adjacent control volumes and, therefore, the local Riemann problem is bypassed. The kth-order spatial derivatives are updated by a central difference of the (k−1)th-order spatial derivatives at cell vertices. All the space-time information is calculated by the Cauchy–Kovalewski procedure. By doing so, the schemes are able to achieve arbitrarily uniform space-time high-order on a compact stencil consisting of only neighboring cells with only one explicit time step. In order to capture discontinuities without spurious oscillations, a weighted essentially non-oscillatory type limiter is tailor-made for the schemes. The limiter preserves the compactness and high-order accuracy of the schemes. The schemes' accuracy, robustness, and efficiency are verified by several numerical examples of scalar conservation laws and the compressible Euler equations.

High Order Positivity- and Bound-Preserving Hybrid Compact-WENO Finite Difference Scheme for the Compressible Euler Equations

Based on the same hybridization framework of Don et al. (SIAM J Sci Comput 38:A691–A711 2016), an improved hybrid scheme employing the nonlinear 5th-order characteristic-wise WENO-Z5 finite difference scheme for capturing high gradients and discontinuities in an essentially non-oscillatory manner and the linear 5th-order conservative compact upwind (CUW5) scheme for resolving the fine scale structures in the smooth regions of the solution in an efficient and accurate manner is developed. By replacing the 6th-order non-dissipative compact central scheme (CCD6) with the CUW5 scheme, which has a build-in dissipation, there is no need to employ an extra high order smoothing procedure to mitigate any numerical oscillations that might appear in an hybrid scheme. The high order multi-resolution algorithm of Harten is employed to detect the smoothness of the solution. To handle the problems with extreme conditions, such as high pressure and density ratios and near vacuum states, and detonation diffraction problems, we design a positivity- and bound-preserving limiter by extending the one developed in Hu et al. (J Comput Phys 242, 2013) for solving the high Mach number jet flows, detonation diffraction problems and detonation passing multiple obstacles problems. Extensive one- and two-dimensional shocked flow problems demonstrate that the new hybrid scheme is less dispersive and less dissipative, and allows a potential speedup up to a factor of more than one and half times faster than the WENO-Z5 scheme.

Prediction of cavitating flow noise by direct numerical simulation

In this study, a direct numerical simulation procedure for the cavitating flow noise is presented. The compressible Navier–Stokes equations are written for the two-phase fluid, employing a density-based homogeneous equilibrium model with a linearly-combined equation of state. To resolve the linear and non-linear waves in the cavitating flow, a sixth-order compact central scheme is utilized with the selective spatial filtering technique. The present cavitation model and numerical methods are validated for two benchmark problems: linear wave convection and acoustic saturation in a bubbly flow. The cavitating flow noise is then computed for a 2D circular cylinder flow at Reynolds number based on a cylinder diameter, 200 and cavitation numbers, σ = 0.7 – 2 . It is observed that, at cavitation numbers σ = 1 and 0.7, the cavitating flow and noise characteristics are significantly changed by the shock waves due to the coherent collapse of the cloud cavitation in the wake. To verify the present direct simulation and further analyze the sources of cavitation noise, an acoustic analogy based on a classical theory of Fitzpatrik and Strasberg is derived. The far-field noise predicted by direct simulation is well compared with that of acoustic analogy, and it also confirms the f - 2 decaying rate in the spectrum, as predicted by the model of Fitzpatrik and Strasberg with the Rayleigh–Plesset equation.

Finite Volume Formulation of Compact Upwind and Central Schemes with Artificial Selective Damping

The present paper describes the use of compact upwind and compact central schemes in a Finite Volume formulation with an extension towards arbitrary meshes. The different schemes are analyzed and tested on several numerical experiments. A new formulation of artificial selective damping that is applicable on non-uniform Cartesian meshes is presented. Results are shown for a 1D advection equation, a 2D rotating Gaussian pulse and a subsonic inviscid vortical flow on uniform and non-uniform meshes and for a non-linear acoustic pulse.

A finite volume formulation of compact central schemes on arbitrary structured grids

In the present paper a formulation allowing the use of compact schemes in the finite volume context on arbitrary meshes is presented. The proposed formulation is based on the use of an implicit formula to evaluate the fluxes on the cell faces. A series of numerical experiments for a 2D model convection equation, a flat plate, a subsonic vortical problem as well as the LES simulation of channel flow has been carried out. The results indicate an important improvement in obtained accuracy compared to a standard central finite volume formulation.