CIP/multi-moment finite volume method for Euler equations: A semi-Lagrangian characteristic formulation

Abstract

An accurate algorithm for the hyperbolic equations has been proposed by combining the constrained interpolation profile/multi-moment finite volume method (CIP/MM FVM) with the characteristic theory. Two types of moments, i.e. the point value (PV) at cell boundary of each mesh element and the volume-integrated average (VIA) over each mesh cell of a physical field, are treated as the model variables and updated independently in time. The interpolation that uses both PV and VIA is reconstructed for each Riemann invariant of the hyperbolic conservation laws. The PVs are then updated by semi-Lagrangian schemes along the characteristic curves, while the VIAs are computed by formulations of flux form, where the numerical fluxes are evaluated by averaging the physical fields over the characteristic curves. The Runge–Kutta type schemes are used for integrating the trajectory equations based on the characteristic speeds to improve the accuracy in time.

The numerical procedure for the one-dimensional Euler conservation laws is described in detail in this paper. Number of benchmark tests are presented. The numerical results show that the present method is accurate and competitive to other existing methods.

Introduction

Many high resolution finite volume schemes have been so far developed for the Euler conservation laws that have direct applications in aerodynamic engineering. A finite volume method (FVM) is cast in a flux form and thus guarantees the numerical conservation which is found to be essential to capturing shock waves with correct position. In a conventional finite volume method, the discretized model variable of physical field is usually defined as the volume-integrated average (VIA), or cell-integrated average, over the control volume. The numerical flux on each control volume boundary is approximated by an interpolation reconstruction based on the VIA for each physical field. A high resolution scheme requires the reconstruction to be built in such a way so that the numerical solution has an accuracy higher than second-order for the smooth region, but does not produce significant numerical oscillations in the presence of the discontinuities. Representative schemes of this category are the monotone upwind scheme for conservation laws (MUSCL) [21], the total variation diminishing (TVD) [6], the piecewise parabolic method (PPM) [1], the piecewise rational method (PRM) [35], the essentially non-oscillatory (ENO) [7], [17], [18], and the weighted ENO (WENO) [11], [9]. In these methods, only the VIA is used as the model variable which is stored and predicted in time. Thus, a high order reconstruction needs a wide stencil of computation grid, which makes the extension of such a method to unstructured grid not a easy task.

Different from the above-mentioned conventional conservative high resolution schemes, where only the VIA of each physical field is treated as the model variable to be put forward in time, the discontinuous Galerkin (DG) method [2], [3], [4], [5] makes use of some extra moments in addition to the VIA-equivalent quantity as the model variables too. The VIA-equivalent moment in a DG method is effectively updated through a finite volume formulation of flux form, and therefore is numerically conserved. A high-order reconstruction can be built with high-order basis functions in a DG method by increasing the local degrees of freedom (DOF) within each control volume (mesh element). Each moment (or DOF) in a DG method has its own evolutionary governing equation derived from the Galerkin formulation, which involves numerical quadratures. Another way to increase local DOF is found in the spectral finite volume (SV) method [23], [24], [25], [26], where each mesh cell or spectral volume (SV) is subdivided into smaller regions, namely control volumes (CVs). The VIA of physical field is then defined over each CV and updated separately in time. So, high-order polynomials can be piecewisely constructed over each SV, and the Riemann problems among the CV of the same SV can be solved exactly. A recent study [23], however, showed that more restrictive CFL condition for computational stability is required for higher order approximations of either DG or SV method.

An alternative to increase the local DOF is found in the constrained interpolation profile (CIP) method [36], [38], where more than two types of moments, i.e. the point value (PV) and the first-order derivatives (DV) are simultaneously treated as the model variables and predicted independently in time. Successive studies have resulted in a class of conservative schemes [20], [37], [28], [29], so called CIP-conservative semi-Lagrangian (CIP-CSL) methods, for the scalar conservative advection transport. In a CIP-CSL scheme, a moment of VIA is introduced as a new model variable that is updated by a flux-form formulation and exactly conserved. More recently, a more general finite volume framework, the CIP/multi-moment finite volume method (CIP/MM FVM), has been proposed and implemented to various fluid dynamic simulations [30], [31], [32], [33], [34]. Using multi-moment, a CIP/MM FVM can construct the high-order interpolation function on a local base, which makes the implementation of the method on unstructured mesh much easier. We, for example, have devised a fourth-order and single-cell based advection scheme on triangular unstructured mesh [8].

A similar multi-moment concept has been used in the Hermite WENO (HWENO) scheme [12] that makes the stencil much more compact than the conventional WENO scheme. In general, making use of multi-moments enables one to reconstruct high-order interpolation function with local stencil.

In the CIP/MM FVM, the way to update the moment is more ‘physically motivated’ and more flexible. In regard to the hyperbolic conservation systems, for example, we compute the VIA (conservative moment) through a flux-form formulation to achieve the numerical conservativeness, and update the PV (non-conservative moment) by a semi-Lagrangian procedure. As shown in [8], a CIP/MM FVM works well even with a larger CFL number. Concerning the computation of the Euler equations, a CIP/MM FVM has been developed by using a pressure projection [32], [33]. The resulting algorithm conserves numerically the VIA of the conservative variables in the Euler equations, and works for all Mach numbers.

In this paper, we present another more accurate variant of the CIP/MM FVM to the one-dimensional inviscid Euler conservation laws by fully implementing the characteristic theory to the system. The semi-Lagrangian solutions are solved in terms of the Riemann invariants along the characteristic curves, and used also to evaluate the numerical fluxes for updating the VIAs of the conservative variables. It results in a robust and accurate formulation for the Euler conservation laws.

In Section 2, we describe the CIP/MM FVM for the scalar conservation law, where the definition for the moments, the CIP-CSL reconstructions and the way to update the moments are given in detail. The implementation of the CIP/MM FVM to the one-dimensional Euler conservation laws is discussed in Section 3, and the numerical experiments for the Euler conservation laws are shown in Section 4. We evaluated our method with some typical benchmark tests. Finally, some discussions and conclusion remarks end the paper in Section 5.