A multislope MUSCL method on unstructured meshes applied to compressible Euler equations for axisymmetric swirling flows

A finite volume method for the numerical solution of axisymmetric inviscid swirling flows is presented. The governing equations of the flow are the axisymmetric compressible Euler equations including swirl (or tangential) velocity. A first-order scheme is introduced where the convective fluxes at cell interfaces are evaluated by the Rusanov or the HLLC numerical flux while the geometric source terms are discretizated to provide a well-balanced scheme i.e. the steady-state solutions with null velocity are preserved. Extension to the second-order space approximation using a multislope MUSCL method is then derived. To test the numerical scheme, a stationary solution of the fluid flow following the radial direction has been established with a zero and nonzero tangential velocity. Numerical and exact solutions are compared for classical Riemann problems where we employ different limiters and effectiveness of the multislope MUSCL scheme is demonstrated for strongly shocked axially symmetric flows like in spherical bubble compression problem. Two other tests with axisymmetric geometries are performed: the supersonic flow in a tube with a cone and the axisymmetric blunt body with a free stream.     

L-stability of vertex-based MUSCL finite volume schemes on unstructured grids: Simulation of incompressible flows with high density ratios

This work is devoted to the design of multi-dimensional finite volume schemes for solving transport equations on unstructured grids. In the framework of MUSCL vertex-based methods we construct numerical fluxes such that the local maximum property is guaranteed under an explicit Courant–Friedrichs–Levy condition. The method can be naturally completed by adaptive local mesh refinements and it turns out that the mesh generation is less constrained than when using the competitive cell-centered methods. We illustrate the effectiveness of the scheme by simulating variable density incompressible viscous flows. Numerical simulations underline the theoretical predictions and succeed in the computation of high density ratio phenomena such as a water bubble falling in air.     

Development and comparison of different spatial numerical schemes for the radiative transfer equation resolution using three-dimensional unstructured meshes

In the present work four different spatial numerical schemes have been developed with the aim of reducing the false-scattering of the numerical solutions obtained with the discrete ordinates (DOM) and the finite volume (FVM) methods. These schemes have been designed specifically for unstructured meshes by means of the extrapolation of nodal values of intensity on the studied radiative direction. The schemes have been tested and compared in several 3D benchmark test cases using both structured orthogonal and unstructured grids.     

Hierarchical reconstruction for spectral volume method on unstructured grids

The hierarchical reconstruction (HR) [Y.-J. Liu, C.-W. Shu, E. Tadmor, M.-P. Zhang, Central discontinuous Galerkin methods on overlapping cells with a non-oscillatory hierarchical reconstruction, SIAM J. Numer. Anal. 45 (2007) 2442–2467; Z.-L. Xu, Y.-J. Liu, C.-W. Shu, Hierarchical reconstruction for discontinuous Galerkin methods on unstructured grids with a WENO type linear reconstruction and partial neighboring cells, J. Comput. Phys. 228 (2009) 2194–2212] is applied to a piecewise quadratic spectral volume method on two-dimensional unstructured grids as a limiting procedure to prevent spurious oscillations in numerical solutions. The key features of this HR are that the reconstruction on each control volume only uses adjacent control volumes, which forms a compact stencil set, and there is no truncation of higher degree terms of the polynomial. We explore a WENO-type linear reconstruction on each hierarchical level for the reconstruction of high degree polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed. We demonstrate that the hierarchical reconstruction can generate essentially non-oscillatory solutions while keeping the resolution and desired order of accuracy for smooth solutions.     

Compact third-order limiter functions for finite volume methods

We consider finite volume methods for the numerical solution of conservation laws. In order to achieve high-order accurate numerical approximation to non-linear smooth functions, we introduce a new class of limiter functions for the spatial reconstruction of hyperbolic equations. We therefore employ and generalize the idea of double-logarithmic reconstruction of Artebrant and Schroll [R. Artebrant, H.J. Schroll, Limiter-free third order logarithmic reconstruction, SIAM J. Sci. Comput. 28 (2006) 359-381].     

A multi-moment finite volume formulation for shallow water equations on unstructured mesh

A novel and accurate finite volume method has been presented to solve the shallow water equations on unstructured grid in plane geometry. In addition to the volume integrated average (VIA moment) for each mesh cell, the point values (PV moment) defined on cell boundary are also treated as the model variables. The volume integrated average is updated via a finite volume formulation, and thus is numerically conserved, while the point value is computed by a point-wise Riemann solver. The cell-wise local interpolation reconstruction is built based on both the VIA and the PV moments, which results in a scheme of almost third order accuracy. Efforts have also been made to formulate the source term of the bottom topography in a way to balance the numerical flux function to satisfy the so-called C-property. The proposed numerical model is validated by numerical tests in comparison with other methods reported in the literature.     

Mesh quality effects on the accuracy of CFD solutions on unstructured meshes

The order of accuracy and error magnitude of node- and cell-centered schemes are examined on representative unstructured meshes and flowfield solutions for computational fluid dynamics. Specifically, we investigate the properties of inviscid and viscous flux discretizations for isotropic and highly stretched meshes using the Method of Manufactured Solutions. Grid quality effects are studied by randomly perturbing the base meshes and cataloguing the error convergence as a function of grid size. For isotropic grids, node-centered approaches produce less error than cell-centered approaches. Moreover, a corrected node-centered scheme is shown to maintain third order accuracy for the inviscid terms on arbitrary triangular meshes. In contrast, for stretched meshes, cell-centered schemes are favored, with cell-centered prismatic approaches in particular showing the lowest levels of error. In three dimensions, simple flux integrations on non-planar control volume faces lead to first-order solution errors, while second-order accuracy is recovered by triangulation of the non-planar faces.     

Filtered density function simulator on unstructured meshes

A new computational filtered density function (FDF) methodology is developed for large eddy simulation (LES) of turbulent reacting flows. This methodology is based on a Lagrangian Monte Carlo (MC) FDF solver constructed on a domain portrayed by an unstructured mesh. The base filtered transport equations on this mesh are solved by a finite-volume (FV) method. The consistency of the hybrid FV-MC solver and the realizability of the simulated results are demonstrated via LES of a temporally developing mixing layer. The overall performance of the model is appraised by comparison with direct numerical simulation (DNS) data. The algorithmic implementation in the commercial software ANSYS-FLUENT facilitates future FDF-LES of turbulent combustion in complex configurations.     

A third-order finite-volume residual-based scheme for the 2D Euler equations on unstructured grids

A residual-based (RB) scheme relies on the vanishing of residual at the steady-state to design a transient first-order dissipation, which becomes high-order at steady-state. Initially designed within a finite-difference framework for computations of compressible flows on structured grids, the RB schemes displayed good convergence, accuracy and shock-capturing properties which motivated their extension to unstructured grids using a finite volume (FV) method. A second-order formulation of the FV–RB scheme for compressible flows on general unstructured grids was presented in a previous paper. The present paper describes the derivation of a third-order FV–RB scheme and its application to hyperbolic model problems as well as subsonic, transonic and supersonic internal and external inviscid flows.     

Arbitrary Lagrangian Eulerian formulation for two-dimensional flows using dynamic meshes with edge swapping

The dynamic modification of the computational grid due to element displacement, deformation and edge swapping is described here in terms of suitably-defined continuous (in time) alterations of the geometry of the elements of the dual mesh. This new interpretation allows one to describe all mesh modifications within the arbitrary Lagrangian Eulerian framework, thus removing the need to interpolate the solution across computational meshes with different connectivity. The resulting scheme is by construction conservative and it is applied here to the solution of the Euler equations for compressible flows in two spatial dimensions. Preliminary two dimensional numerical simulations are presented to demonstrate the soundness of the approach. Numerical experiments show that this method allows for large time steps without causing element invalidation or tangling and at the same time guarantees high quality of the mesh elements without resorting to global re-meshing techniques, resulting in a very efficient solver for the analysis of e.g. fluid-structure interaction problems, even for those cases that require large mesh deformations or changes in the domain topology.     

Accuracy preserving limiter for the high-order accurate solution of the Euler equations

Higher-order finite-volume methods have been shown to be more efficient than second-order methods. However, no consensus has been reached on how to eliminate the oscillations caused by solution discontinuities. Essentially non-oscillatory (ENO) schemes provide a solution but are computationally expensive to implement and may not converge well for steady-state problems. This work studies the extension of limiters used for second-order methods to the higher-order case. Requirements for accuracy and efficient convergence are discussed. A new limiting procedure is proposed. Ringleb’s flow problem is used to demonstrate that nearly nominal orders of accuracy for schemes up to fourth-order can be achieved in smooth regions using the new limiter. Results for the fourth-order accurate solution of transonic flow demonstrates good convergence properties and significant qualitative improvement of the solution relative the second-order method. The new limiter can also be successfully applied to reduce the dissipation of second-order schemes with minimal sacrifices in convergence properties relative to existing approaches.     

On positivity-preserving high order discontinuous Galerkin schemes for compressible Euler equations on rectangular meshes

We construct uniformly high order accurate discontinuous Galerkin (DG) schemes which preserve positivity of density and pressure for Euler equations of compressible gas dynamics. The same framework also applies to high order accurate finite volume (e.g. essentially non-oscillatory (ENO) or weighted ENO (WENO)) schemes. Motivated by Perthame and Shu (1996) [20] and Zhang and Shu (2010) [26], a general framework, for arbitrary order of accuracy, is established to construct a positivity preserving limiter for the finite volume and DG methods with first order Euler forward time discretization solving one-dimensional compressible Euler equations. The limiter can be proven to maintain high order accuracy and is easy to implement. Strong stability preserving (SSP) high order time discretizations will keep the positivity property. Following the idea in Zhang and Shu (2010) [26], we extend this framework to higher dimensions on rectangular meshes in a straightforward way. Numerical tests for the third order DG method are reported to demonstrate the effectiveness of the methods.     

Point-wise hierarchical reconstruction for discontinuous Galerkin and finite volume methods for solving conservation laws

We develop a new hierarchical reconstruction (HR) method  and  for limiting solutions of the discontinuous Galerkin and finite volume methods up to fourth order of accuracy without local characteristic decomposition for solving hyperbolic nonlinear conservation laws on triangular meshes. The new HR utilizes a set of point values when evaluating polynomials and remainders on neighboring cells, extending the technique introduced in Hu, Li and Tang [9]. The point-wise HR simplifies the implementation of the previous HR method which requires integration over neighboring cells and makes HR easier to extend to arbitrary meshes. We prove that the new point-wise HR method keeps the order of accuracy of the approximation polynomials. Numerical computations for scalar and system of nonlinear hyperbolic equations are performed on two-dimensional triangular meshes. We demonstrate that the new hierarchical reconstruction generates essentially non-oscillatory solutions for schemes up to fourth order on triangular meshes.     

A two-dimensional unstructured cell-centered multi-material ALE scheme using VOF interface reconstruction

We present a new cell-centered multi-material arbitrary Lagrangian–Eulerian (ALE) scheme to solve the compressible gas dynamics equations on two-dimensional unstructured grid. Our ALE method is of the explicit time-marching Lagrange plus remap type. Namely, it involves the following three phases: a Lagrangian phase wherein the flow is advanced using a cell-centered scheme; a rezone phase in which the nodes of the computational grid are moved to more optimal positions; a cell-centered remap phase which consists of interpolating conservatively the Lagrangian solution onto the rezoned grid. The multi-material modeling utilizes either concentration equations for miscible fluids or the Volume Of Fluid (VOF) capability with interface reconstruction for immiscible fluids. The main original feature of this ALE scheme lies in the introduction of a new mesh relaxation procedure which keeps the rezoned grid as close as possible to the Lagrangian one. In this formalism, the rezoned grid is defined as a convex combination between the Lagrangian grid and the grid resulting from condition number smoothing. This convex combination is constructed through the use of a scalar parameter which is a scalar function of the invariants of the Cauchy–Green tensor over the Lagrangian phase. Regarding the cell-centered remap phase, we employ two classical methods based on a partition of the rezoned cell in terms of its overlap with the Lagrangian cells. The first one is a simplified swept face-based method whereas the second one is a cell-intersection-based method. Our multi-material ALE methodology is assessed through several demanding two-dimensional tests. The corresponding numerical results provide a clear evidence of the robustness and the accuracy of this new scheme.     

A conservative level set method suitable for variable-order approximations and unstructured meshes

This paper presents a formulation for free-surface computations capable of handling complex phenomena, such as wave breaking, without excessive mass loss or smearing of the interface. The formulation is suitable for discretizations using finite elements of any topology and order, or other approaches such as isogeometric and finite volume methods. Furthermore, the approach builds on standard level set tools and can therefore be used to augment existing implementations of level set methods with discrete conservation properties. Implementations of the method are tested on several difficult two- and three-dimensional problems, including two incompressible air/water flow problems with available experimental results. Linear and quadratic approximations on unstructured tetrahedral and trilinear approximations on hexahedral meshes were tested. Global conservation and agreement with experiments as well as computations by other researchers are obtained.     

A unifying lifting collocation penalty formulation including the discontinuous Galerkin, spectral volume/difference methods for conservation laws on mixed grids

Recently a new high-order formulation for 1D conservation laws was developed by Huynh using the idea of “flux reconstruction”. The formulation was capable of unifying several popular methods including the discontinuous Galerkin, staggered-grid multi-domain method, or the spectral difference/spectral volume methods into a single family. The extension of the method to quadrilateral and hexahedral elements is straightforward. In an attempt to extend the method to other element types such as triangular, tetrahedral or prismatic elements, the idea of “flux reconstruction” is generalized into a “lifting collocation penalty” approach. With a judicious selection of solution points and flux points, the approach can be made simple and efficient to implement for mixed grids. In addition, the formulation includes the discontinuous Galerkin, spectral volume and spectral difference methods as special cases. Several test problems are presented to demonstrate the capability of the method.     

A monotone finite volume method for advection–diffusion equations on unstructured polygonal meshes

We present a new second-order accurate monotone finite volume (FV) method for the steady-state advection–diffusion equation. The method uses a nonlinear approximation for both diffusive and advective fluxes and guarantees solution non-negativity. The interpolation-free approximation of the diffusive flux uses the nonlinear two-point stencil proposed in Lipnikov [23]. Approximation of the advective flux is based on the second-order upwind method with a specially designed minimal nonlinear correction. The second-order convergence rate and monotonicity are verified with numerical experiments.     

High order multi-moment constrained finite volume method. Part I: Basic formulation

A new high-order finite volume method based on local reconstruction is presented in this paper. The method, so-called the multi-moment constrained finite volume (MCV) method, uses the point values defined within single cell at equally spaced points as the model variables (or unknowns). The time evolution equations used to update the unknowns are derived from a set of constraint conditions imposed on multi kinds of moments, i.e. the cell-averaged value and the point-wise value of the state variable and its derivatives. The finite volume constraint on the cell-average guarantees the numerical conservativeness of the method. Most constraint conditions are imposed on the cell boundaries, where the numerical flux and its derivatives are solved as general Riemann problems. A multi-moment constrained Lagrange interpolation reconstruction for the demanded order of accuracy is constructed over single cell and converts the evolution equations of the moments to those of the unknowns. The presented method provides a general framework to construct efficient schemes of high orders. The basic formulations for hyperbolic conservation laws in 1- and 2D structured grids are detailed with the numerical results of widely used benchmark tests.     

A high-order finite volume method for systems of conservation laws—Multi-dimensional Optimal Order Detection (MOOD)

In this paper, we investigate an original way to deal with the problems generated by the limitation process of high-order finite volume methods based on polynomial reconstructions. Multi-dimensional Optimal Order Detection (MOOD) breaks away from classical limitations employed in high-order methods. The proposed method consists of detecting problematic situations after each time update of the solution and of reducing the local polynomial degree before recomputing the solution. As multi-dimensional MUSCL methods, the concept is simple and independent of mesh structure. Moreover MOOD is able to take physical constraints such as density and pressure positivity into account through an “a posteriori” detection. Numerical results on classical and demanding test cases for advection and Euler system are presented on quadrangular meshes to support the promising potential of this approach.