A Scale-Invariant Fifth Order WCNS Scheme for Hyperbolic Conservation Laws

In this article,a robust,effective,and scale-invariant weighted compact nonlinear scheme(WCNS)is proposed by introducing descaling techniques to the nonlinear weights of the WCNS-Z/D schemes.The new scheme achieves an essentially non-oscillatory approximation of a discontinuous function(ENO-property),a scaleinvariant property with an arbitrary scale of a function(Si-property),and an optimal order of accuracy with smooth function regardless of the critical point(Cp-property).The classical WCNS-Z/D schemes do not satisfy Si-property intrinsically,which is caused by a loss of sub-stencils’adaptivity in the nonlinear interpolation of a discontinuous function when scaled by a small scale factor.A new nonlinear weight is devised by using an average of the function values and the descaling function,providing the new WCNS schemes(WCNS-Zm/Dm)with many attractive properties.The ENO-property,Si-property and Cp-property of the new WCNS schemes are validated numerically.Results show that the WCNS-Zm/Dm schemes satisfy the ENO-property and Si-property,while only the WCNS-Dm scheme satisfies the Cp-property.In addition,the Gaussian wave problem is solved by using successively refined grids to verify that the optimal order of accuracy of the new schemes can be achieved.Several one-dimensional shock tube problems,and two-dimensional double Mach reflection(DMR)problem and the Riemann IVP problem are simulated to illustrate the ENOproperty and Si-property of the scale-invariant WCNS-Zm/Dm schemes.

Extending the global-direction stencil with“face-area-weighted centroid”to unstructured finite volume discretization from integral formfinitevolumediscretization from integral form

Accuracy of unstructured finite volume discretization is greatly influenced by the gradient reconstruction.For the commonly used k-exact reconstruction method,the cell centroid is always chosen as the reference point to formulate the reconstructed function.But in some practical problems,such as the boundary layer,cells in this area are always set with high aspect ratio to improve the local field resolution,and if geometric centroid is still utilized for the spatial discretization,the severe grid skewness cannot be avoided,which is adverse to the numerical performance of unstructured finite volume solver.In previous work[Kong,et al.Chin Phys B 29(10):100203,2020],we explored a novel global-direction stencil and combined it with the face-area-weighted centroid on unstructured finite volume methods from differential form to realize the skewness reduction and a better reflection of flow anisotropy.Greatly inspired by the differential form,in this research,we demonstrate that it is also feasible to extend this novel method to the unstructured finite volume discretization from integral form on both second and third-order finite volume solver.Numerical examples governed by linear convective,Euler and Laplacian equations are utilized to examine the correctness as well as effectiveness of this extension.Compared with traditional vertex-neighbor and face-neighbor stencils based on the geometric centroid,the grid skewness is almost eliminated and computational accuracy as well as convergence rate is greatly improved by the global-direction stencil with face-area-weighted centroid.As a result,on unstructured finite volume discretization from integral form,the method also has superiorities on both computational accuracy and convergence rate.