Construction Methodology of Weighted Upwind Compact Scheme

Abstract

A methodology from dissipation and dispersion perspective is introduced for constructing Weighted Upwind Compact Scheme in this paper. This construction methodology can derive a parameter weighted upwind compact scheme not only has high order accuracy in smooth area or for smooth function, but also the capability of shock-capturing. This construction methodology utilizes compact scheme to present three stencils with parameters as coefficients of function values and derivatives. Initially, these parameters can keep each stencil with third order accuracy and keep the combined scheme with seventh order accuracy. Then by giving dissipation and dispersion constraints to left stencil, right stencil and the combined scheme, those parameters from the initial step can be found. Some constrains are added to prevent the dissipation from negative value for the upwind and central stencils which is the key issue to avoid non-physical oscillations.

Matrix Form of Deriving High Order Schemes for the First Derivative

“…It is well known (4)(5)(6) that implicit schemes are significantly more accurate for the small scales than explicit schemes with the same stencil width, i.e., the number of nodes in the scheme. Many implicit or spectral-like compact schemes have been developed (4,(7)(8)(9). Since the spectral methods require the flow to be simple in both domain and boundary condition, many attempts have been made to overcome this drawback of the spectral methods (9)(10).…”

“…Many implicit or spectral-like compact schemes have been developed (4,(7)(8)(9). Since the spectral methods require the flow to be simple in both domain and boundary condition, many attempts have been made to overcome this drawback of the spectral methods (9)(10).…”