Fifth order finite volume WENO in general orthogonally - curvilinear coordinates

Abstract

High order reconstruction in the finite volume (FV) approach is achieved by a more fundamental form of the fifth order WENO reconstruction in the framework of orthogonally−curvilinear coordinates, for solving hyperbolic conservation equations. The derivation employs a piecewise parabolic polynomial approximation to the zone averaged values (Q i ) to reconstruct the right (q + i ), middle (q M i ), and left (q − i ) interface values. The grid dependent linear weights of the WENO are recovered by inverting a Vandermonde−like linear system of equations with spatially varying coefficients. A scheme for calculating the linear weights, optimal weights, and smoothness indicator on a regularly−/irregularly−spaced grid in orthogonally−curvilinear coordinates is proposed. A grid independent relation for evaluating the smoothness indicator is derived from the basic definition.Finally, a computationally efficient extension to multi-dimensions is proposed along with the procedures for flux and source term integrations. Analytical values of the linear weights, optimal weights, and weights for flux and source term integrations are provided for a regularly−spaced grid in Cartesian, cylindrical, and spherical coordinates. Conventional fifth order WENO−JS can be fully recovered in the case of limiting curvature (R → ∞).The fifth order finite volume WENO−C (orthogonally−curvilinear version of WENO) reconstruction scheme is tested for several 1D and 2D benchmark tests involving smooth and discontinuous flows in cylindrical and spherical coordinates.including discontinuous flows [7,8], smooth flows with turbulence [9] [10], aeroacoustics [10], sediment transport [11] and magnetohydrodynamics (MHD) [12,13,14]. In a plethora of reconstruction techniques including p th order accurate essentially non−oscillatory (ENO) scheme [15], second order total variation diminishing (TVD) methods [2], discontinuous Galerkin methods [10], and modified piecewise parabolic method (PPM) [2,16,17,18], WENO stands a chance by its virtue of attaining a convexly combined (2p − 1) th order of convergence for smooth flows aided with a novel ENO strategy for maintaining high order accuracy even for the discontinuous flows [2,15].The conventional WENO scheme is specifically designed for the reconstruction in Cartesian coordinates on uniform grids [4,5]. For an arbitrary curvilinear mesh, the procedure of using a Jacobian, in order to map a general curvilinear mesh to a uniform Cartesian mesh, is employed [15]. However, the employment of Cartesianbased reconstruction scheme on a curvilinear grid suffers from a number of drawbacks, e.g., in the original PPM paper [16], reconstruction was performed in volume coordinates (than the linear ones) so that algorithm for a Cartesian mesh can be used on a cylindrical/spherical mesh. However, the resulting interface states became first order accurate even for smooth flows [16]. Another example can be the volume average assignment to the geometrical cell center of finite volume than the centroid [19,20,21]. The reconstructi...

Fifth-Order Finite-Volume WENO on Cylindrical Grids

“…(4), which is the fundamental equation for reconstruction in cylindrical coordinates. For the detailed derivation, kindly refer to [3].…”

“…Also, they conform to the original WENO-JS [1] for the limiting case (R → ∞). The weights required for source term and flux integration in one or more dimensions are given in [3].…”

“…The nonlinear weight (ω ± i,l ) for the WENO-C interpolation is defined in Eq. (8) [1], where is chosen to be 10 −6 [1,3].…”

“…A smooth problem involving a nonlinear system of 1D gas dynamical equations is solved to test fifth-order accuracy of the spatial discretization scheme [3]. The Euler equations in cylindrical-radial coordinates can be written in the form of Eq.…”

“…Sedov explosion test is performed to investigate code's ability to deal with strong shocks and non-planar symmetry [3]. The problem involves a self-similar evolution of a cylindrical blastwave in a uniform grid (N = 100) from a localized initial pressure perturbation (delta-function) in an otherwise homogeneous medium.…”