Explicit and Implicit Multidimensional Compact High-Resolution Shock-Capturing Methods:Formulation

✍ Scribed by H.C. Yee

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 441 KB
• Volume: 131
• Category: Article
• ISSN: 0021-9991
• DOI: 10.1006/jcph.1996.5608

✦ Synopsis

Two families of explicit and implicit compact high-resolution shock-capturing methods for the multidimensional compressible pressible shear-layer flow and direct numerical simulation Euler equations for fluid dynamics are constructed. Some of these of turbulence. The reader is referred to [2][3] for more schemes can be fourth-and sixth-order accurate away from discontidetails. The papers by Lele and Davis discuss wave resolunuities. For the semi-discrete case their shock-capturing properties tion and phase errors for linear wave propagation. Alare of the total variation diminishing (TVD), total variation bounded though formal extension of their schemes to nonlinear (TVB), total variation diminishing in the mean (TVDM), essentially nonoscillatory (ENO), or positive type of scheme for 1D scalar hypersystems is a straightforward, systematic extension of their bolic conservation laws and are positive schemes in more than one idea to minimize phase errors and enhance wave resolution dimension. These higher-order compact schemes require the same for coupled nonlinear systems of equations remains to be grid stencil per spatial direction as their second-order noncompact seen. Unlike the standard compact schemes that use symcousins. The added terms over the second-order noncompact cousins involve extra vector additions but no added flux evaluations. metric compact operators, most of the recent development Due to the construction, these schemes can be viewed as approxiin compact methods uses asymmetric compact operators.

mations to genuinely multidimensional schemes in the sense that They also require additional numerical dissipation for high they might produce less distortion in spherical type shocks and are gradient flows and generate spurious oscillations across more accurate in vortex type flows than schemes based purely on shock waves and contact discontinuities even with added 1D extensions. The extension of these families of compact schemes to coupled nonlinear systems can be accomplished using the Roe linear numerical dissipation. At present, there is no systemapproximate Riemann solver, the generalized Steger and Warming atic extension of these asymmetric compact schemes to have flux-vector splitting, or the van Leer type flux-vector splitting. Modihigh-resolution shock-capturing capability. Hybrid fication to existing high-resolution second-or third-order nonschemes using these types of compact methods in conjunccompact shock-capturing computer codes is minimal. High-resolution with completely different construction of high-resolution shock-capturing properties can also be achieved via a variant of the second-order Lax-Friedrichs numerical flux without the use tion shock-capturing methods to enhance shock resolution of Riemann solvers for coupled nonlinear systems with comparable were also proposed (see, e.g., ). A shortcoming of this operations count to their classical shock-capturing counterparts. An type of hybridization is that the numerical solution might efficient and compatible high-resolution shock-capturing filter for experience a nonsmooth transition at the switch to a differspatially fourth-and sixth-order classical compact and noncompact schemes is discussed. The simplest extension to viscous flows can ent type of scheme, in addition to being sensitive to the be achieved by using the standard fourth-order compact or nonchoice of the numerical flux or slope limiter. For 2D and 3D compact formula for the viscous terms. ᮊ 1997 Academic Press complex shock wave and contact surface interactions, the switch mechanism can become less trivial.

The motivation of the present work is to construct proposed a similar idea [6] but with different construction Lake Tahoe, Nevada. A Major portion of this paper was published as NASA TM-110364, August 1995. than the present work . Here we define a compact scheme in 216