WLS-ENO: Weighted-least-squares based essentially non-oscillatory schemes for finite volume methods on unstructured meshes

Abstract

ENO (Essentially Non-Oscillatory) and WENO (Weighted Essentially Non-Oscillatory) schemes are widely used high-order schemes for solving partial differential equations (PDEs), especially hyperbolic conservation laws with piecewise smooth solutions. For structured meshes, these techniques can achieve high order accuracy for smooth functions while being non-oscillatory near discontinuities. For unstructured meshes, which are needed for complex geometries, similar schemes are required but they are much more challenging. We propose a new family of non-oscillatory schemes, called WLS-ENO, in the context of solving hyperbolic conservation laws using finite-volume methods over unstructured meshes. WLS-ENO is derived based on Taylor series expansion and solved using a weighted least squares formulation. Unlike other non-oscillatory schemes, the WLS-ENO does not require constructing sub-stencils, and hence it provides more flexible framework and is less sensitive to mesh quality. We present rigorous analysis of the accuracy and stability of WLS-ENO, and present numerical results in 1-D, 2-D, and 3-D for a number of benchmark problems, and also report some comparisons against WENO.

Fifth order finite volume WENO in general orthogonally - curvilinear coordinates

“…For the Cartesian case as shown in Fig. B.14, the maximum CFL number value obtained is 1.44, similar to a previous study [43]. It can be observed respectively from Figs.…”

“…where (41) and (42), a general form of equation for integration from a lower dimension to a higher dimension can be derived, as given by Eq. (43).…”

“…Also, we define the spectrum S of a spatial discretization scheme in Eq. (B.14)[43].S = {−z(θ k ) : θ k ∈ 0, ∆θ,2∆θ, 2π}, ∆θ = 2π∆ξ (B.14) Rescaled spectrum (with maximum stable CFL numberσ = 1.44) and stability domains of fifth−order WENO−C in Cartesian coordinates (m = 0) in a complex plane The stability limit is thus the largest CFL numberσ such that the rescaled spectrumσS lies inside the stability domain S t .σ S ∈ S t (B.15) For the third−order Runge−Kutta scheme, the amplification factor g is given in Eq. (B.16).…”

Fifth-Order Finite-Volume WENO on Cylindrical Grids

“…In this section, we analyze WENO-C scheme for model problems involving smooth flow in 1D cylindrical-radial coordinates, based on a modified von Neumann stability analysis [4]. We consider scalar advection equation (10) in 1D cylindricalradial coordinates.…”

“…where m = 1 for cylindrical-radial coordinates. Using the same approach as given in [4], we can plot the spatial spectrum {S : −z(θ k ) for θ k ∈ [0, 2π ]} and the stability domain S t for TVD-RK order 3. The maximum stable CFL number of this scheme can be computed by finding the largest rescaling parameterσ , so that the rescaled spectrum still lies in the stability domain.…”