Fifth order finite volume WENO in general orthogonally - curvilinear coordinates

doi:10.1016/j.compfluid.2019.06.031

Computers & Fluids

Volume 190, 15 August 2019, Pages 398-424

https://doi.org/10.1016/j.compfluid.2019.06.031 Get rights and content

Highlights

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Proposed fifth order finite volume WENO-C in orthogonally-curvilinear coordinates.
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WENO-Curvilinear reconstruction and averaging weights provided for standard grids.
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Analytical values of weights for Radius  → ∞ case conform to conventional WENO-JS.
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Tests performed in curvilinear coordinates testify for fifth order & ENO properties.

Abstract

A fifth order finite volume WENO reconstruction scheme is proposed 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 $({\bar{Q}}_{i})$ to reconstruct the right ( $q_{i}^{+}$ ), middle ( $q_{i}^{M}$ ), 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.

Introduction

Weighted essentially non - oscillatory (WENO) reconstruction schemes have been considered among popular numerical approaches for solving one - and two - dimensional hyperbolic conservation laws [1], [2], [3], [4], [5], [6], [7]. Finite volume methods deal with the volume averages, which changes only when there is an imbalance of the fluxes across the control volume [2]. Flux evaluation at an interface requires an important task of reconstructing the cell averaged value at the interface [2]. High order reconstruction is preferred for the cases of complex flow phenomena including discontinuous flows [8], [9], smooth flows with turbulence [10], [11], aeroacoustics [11], sediment transport [12] and magnetohydrodynamics (MHD) [13], [14], [15]. There is a plethora of high order reconstruction schemes such as ENO [16], targeted ENO (TENO) [17], total variation diminishing (TVD) [2], discontinuous Galerkin [11], piecewise parabolic method (PPM) [2], [18], [19], [20], spectral schemes [21], and flux reconstruction methods [22]. However, WENO has received attention by its virtue of attaining a convexly combined very high order of convergence for smooth flows aided with ENO strategy for maintaining a high order accuracy even for the discontinuous flows [2], [16]. Therefore, a number of variants of WENO schemes have been proposed for both structured and unstructured grids such as central and compact WENO schemes [23], [24], [25], [26], [27], [28], [29], [30], [31].

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 [16]. However, the employment of Cartesian-based reconstruction scheme on a curvilinear grid suffers from a number of drawbacks, e.g., in the original PPM paper [18], reconstruction was performed in volume coordinates 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 [18]. Another example can be the volume average assignment to the geometrical cell center of finite volume instead of the centroid [32], [33], [34]. The reconstruction in general coordinates can be performed with the aid of two techniques: genuine multi - dimensional reconstruction and dimension - by - dimension reconstruction [16]. Genuine multi - dimensional reconstruction is computationally expensive and highly complicated since it considers all of the finite volumes while constructing the polynomial [16]. A better approach is to perform a dimension - by - dimension form of reconstruction since it consists of less expensive one - dimensional sweeps in every dimension. Moreover, a number of engineering and scientific problems considered in Cartesian, cylindrical, and spherical coordinates can be tackled using regularly - spaced and irregularly - spaced grids.

A breakthrough in the field of high order reconstruction in these coordinates is the application of the Vandermonde - like linear systems of equations with spatially varying coefficients [2]. It is reintroduced in the present work to build a basis for the derivation of the high order WENO schemes. Mignone [2] restricted the work to the usage of the third order WENO approach with the weight functions provided by Yamaleev and Carpenter [35] and did not extend it to multi - dimensions (2D and 3D). In Mignone’s paper [2], modified piecewise parabolic method (PPM₅) of order $\sim 2 - 3$ gave better results when compared with the modified third order WENO. However, the latter reconstruction scheme gave consistent values for all the numerical tests performed. Also, there is a drop of accuracy in the modified third order WENO scheme for discontinuous flow cases [2] when the standard weights derived by Jiang and Shu [4] are used, as they are specifically restricted to the Cartesian grids.

The motivation for the present work is to develop a fifth order finite volume WENO - C reconstruction scheme in orthogonally - curvilinear coordinates for regularly - spaced and irregularly - spaced grids [36], [37]. It is based on the concepts of linear weights by Mignone [2] and optimal weights, smoothness indicators by Jiang and Shu [4]. Also, the present work provides a computationally efficient extension of this scheme to multi - dimensions and deals with the source terms straightforwardly.

The present work is divided into four sections. Section 2 includes the fifth order finite volume WENO - C reconstruction procedure for a regularly - /irregularly - spaced grid in orthogonally - curvilinear coordinates. It is followed by Section 3 in which 1D and 2D numerical benchmark tests involving smooth and discontinuous flows in cylindrical and spherical coordinates are presented. Finally, Section 4 concludes the paper. Appendix at the end is divided into two sections. The first section includes the analytical values of the weights required for WENO - C reconstruction and flux/source term integration for standard uniform grids, whereas the second section includes linear stability analysis of the proposed scheme.

Section snippets

Finite volume discretization in curvilinear coordinates

The scalar conservation law in an orthogonal system of coordinates (x₁, x₂, x₃) having the scale factors h₁, h₂, h₃ and unit vectors $({\hat{e}}_{1}, {\hat{e}}_{2}, {\hat{e}}_{3})$ in the respective directions, is given in Eq. (1). $\frac{\partial Q}{\partial t} + \nabla . F = S$ where Q is the conserved quantity of the fluid, $F = (F_{1}, F_{2}, F_{3})$ is the corresponding flux vector, and S is the source term. The divergence operator is further expressed in the form of Eq. (2). $\nabla . F = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial x_{1}} (h_{2} h_{3} F_{1}) + \frac{\partial}{\partial x_{2}} (h_{1} h_{3} F_{2}) + \frac{\partial}{\partial x_{3}} (h_{1} h_{2} F_{3})]$

Eq. (1) is discretized over a computational domain

Numerical tests

In this section, several tests on scalar and nonlinear system of equations are performed to analyze the performance of the WENO - C reconstruction scheme. The test cases include scalar advection (1D) on regularly - / irregularly - spaced grids, smooth (1D) and discontinuous inviscid flows (1D/2D) governed by a system of nonlinear equations (Euler equations) on regularly - spaced grids in cylindrical and spherical coordinates. For the sake of comparison solely on the grounds of the high order

Conclusions

The fifth order finite volume WENO - C reconstruction scheme provides a more general framework in orthogonally - curvilinear coordinates to achieve high order spatial accuracy with minimal computational cost. Analytical values of linear weights, optimal weights, weights for mid - point interpolation, and flux/source term integration are derived for the standard grids. The proposed reconstruction scheme can be applied to both regularly - spaced and irregularly - spaced grids. A grid independent

Acknowledgements

The current research is supported by Hong Kong Research Grant Council (16207715, 16206617) and National Natural Science Foundation of China (11772281, 91530319).

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