High-order Finite Volume Method Research Articles

Numerical studies of hydrodynamic relativistic astrophysical jet: The modification of vorticity transport equation

In this Letter, we have carried out the two-dimensional numerical simulation of axisymmetric relativistic jet in cylindrical coordinates by employing higher order finite volume method in PLUTO [Mignone et al., “PLUTO: A numerical code for computational astrophysics,” Astrophys. J. Suppl. Ser. 170, 228 (2007)] solver. The modified vorticity transport equation has been proposed for relativistic flow by taking the curl of the momentum equation, which shows significant change in the baroclinic vorticity production term due to relativistic effect. Both mathematical analysis and numerical results show that the vorticity production term due to baroclinic torque is heavily influenced due to the presence of specific enthalpy gradient and square of Lorentz factor gradient in a relativistic fluid flow.

A fourth-order compact finite volume method on unstructured grids for simulation of two-dimensional incompressible flow

This paper introduces a novel and efficient compact reconstruction procedure for high-order finite volume methods applied to unstructured grids. In this procedure, we establish a set of constitutive relations that ensure the continuity of the reconstruction polynomial and its normal derivatives between adjacent elements at control points. The paper delves into the details of the fourth-order compact reconstruction method specifically designed for two-dimensional triangular grids. This method can be considered an extension of the one-dimensional compact scheme and cubic spline interpolation methods to two-dimensional triangular unstructured grids. In the cubic polynomial reconstruction, a two-dimensional cubic polynomial is reconstructed using the cell averages of elements in the standard stencil and the function values at control points. The determination of function values at control points is achieved by adhering to the constraint of normal derivative continuity. This reconstruction is solved using a relaxation iteration method and has been verified to be solvable for triangular grids. Compared to other fourth-order implicit compact polynomial reconstructions, our method requires solving a smaller number of unknowns, which means less computational cost in reconstruction. By applying this method to solve the Poisson equation, the linear convection equation, and incompressible flow benchmarks, it demonstrates that the proposed method exhibits the expected high-order accuracy and performs well in incompressible flow problems.

High-order finite volume methods for solving compressible multicomponent flows

High-order finite volume methods for solving compressible multicomponent flows

High-order finite volume method for solving compressible multicomponent flows with Mie–Grüneisen equation of state

In this paper, we propose a new high-order finite volume method for solving the multicomponent fluids problem with Mie–Grüneisen EOS. Firstly, based on the cell averages of conservative variables, we develop a procedure to reconstruct the cell averages of the primitive variables in a high-order manner. Secondly, the high-order reconstructions employed in computing numerical fluxes are implemented in a characteristic-wise manner to reduce numerical oscillations as much as possible and obtain high-resolution results. Thirdly, advection equation within the governing system is rewritten in a conservative form with a source term to enhance the scheme’s performance. We utilize integration by parts and high-order numerical integration techniques to handle the source terms. Finally, all variables are evolved by using Runge–Kutta time discretization. All steps are carefully designed to maintain the equilibrium of pressure and velocity for the interface-only problem, which is crucial in designing a high-resolution scheme and adapting to more complex multicomponent problems. We have performed extensive numerical tests for both one- and two-dimensional problems to verify our scheme’s high resolution and accuracy.

A shock capturing artificial viscosity scheme in consistent with the compact high-order finite volume methods

This paper presents a shock capturing artificial viscosity scheme for the compact high-order finite volume methods in terms of the variational reconstructions on unstructured grids. The key for the design of the present artificial viscosity is the smoothness indicator, which is based on the concept of interfacial jump integration, measuring the discontinuities of the reconstruction polynomial and its spatial derivatives across a cell interface. Since the variational reconstruction is carried out by minimizing the functional in terms of the interfacial jump integration, the present smoothness indicator gives a discretization-consistent measurement of the smoothness of the flow fields that is sufficiently large in the region near discontinuities, and is in the same order of magnitude as the spatial truncation error of the finite volume scheme in smooth regions. These properties ensure that the newly developed artificial viscosity scheme has the problem-independent capability to suppress non-physical oscillations near discontinuities and preserve the theoretical order of accuracy for smooth flow. The shock capturing capability of the proposed artificial viscosity scheme has been demonstrated by a number of numerical examples confirming its essentially non-oscillatory and high-resolution properties. Additionally, the proposed artificial viscosity scheme exhibits higher computational efficiency than the approach based on a traditional limiter.

Exploring the potential of TENO and WENO schemes for simulating under-resolved turbulent flows in the atmosphere using Euler equations

This paper focuses on the design and analysis of very high-order finite volume methods for the computation of simplified meso- and micro-scale atmospheric flows. In a dry atmosphere, these flows can be represented by the Euler equations with a gravitational source term. Two different approaches are considered here. While one of the approaches is fully conservative for the total energy, the other is formulated in a non-conservative form. The main focus of the paper is to analyze the performance of such models in combination with the traditional WENO reconstruction and the novel TENO reconstruction by examining the spectral properties of these reconstruction methods. The overarching goal is to determine whether the combination of these models and numerical schemes can be used to build an implicit Large Eddy Simulation framework, shedding light on their potential advantages or limitations in representing under-resolved atmospheric processes in the meso- and micro-scales.

KFVM-WENO: A High-order Accurate Kernel-based Finite Volume Method for Compressible Hydrodynamics

This paper presents a fully multidimensional kernel-based reconstruction scheme for finite volume methods applied to systems of hyperbolic conservation laws, with a particular emphasis on the compressible Euler equations. Nonoscillatory reconstruction is achieved through an adaptive-order weighted essentially nonoscillatory (WENO) method cast into a form suited to multidimensional reconstruction. A kernel-based approach inspired by radial basis functions and Gaussian process modeling, which we call kernel-based finite volume method with WENO, is presented here. This approach allows the creation of a scheme of arbitrary order of accuracy with simply defined multidimensional stencils and substencils. Furthermore, the fully multidimensional nature of the reconstruction allows for a more straightforward extension to higher spatial dimensions and removes the need for complicated boundary conditions on intermediate quantities in modified dimension-by-dimension methods. In addition, a new simple yet effective set of reconstruction variables is introduced, which could be useful in existing schemes with little modification. The proposed scheme is applied to a suite of stringent and informative benchmark problems to demonstrate its efficacy and utility. A highly parallel multi-GPU implementation using Kokkos and the message-passing interface is also provided.

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

A New Discretely Divergence-Free Positivity-Preserving High-Order Finite Volume Method for Ideal MHD Equations

Development and Verification of a Higher-Order Computational Fluid Dynamics Solver

Abstract Over the past two decades, higher-order methods have gained much broader use in computational science and engineering as these schemes are often more efficient per degree-of-freedom at achieving a prescribed error tolerance than lower-order methods. During this time, higher-order variants of most discretization schemes, such as finite difference methods, finite volume methods, and finite element methods, have emerged. The finite volume method is arguably the most widely used discretization technique in production-level computational fluid dynamics solvers due to its robustness and conservation properties. However, most finite volume solvers only employ a conventional second-order scheme. To leverage the benefits of higher-order methods, the higher-order finite volume method seems the most natural for those seeking to extend their legacy solvers to higher-order. Nonetheless, ensuring higher-order accuracy is maintained is quite challenging as the implementation requirements for a higher-order scheme are much greater than those of a lower-order scheme. In this work, a methodology for verifying higher-order finite volume codes is presented. The higher-order finite volume method is outlined in detail. Order verification tests are proposed for all major components, including the treatment of curved boundaries and the higher-order solution reconstruction. System-level verification tests are performed using the weak form of the method of manufactured solutions. Several canonical verification cases are also presented for the Euler and laminar Navier–Stokes equations.

Modelling of the water on deck problem for an offshore supply vessel running in regular astern waves

The motion of trapped water on the weather deck of an offshore supply vessel running in regular astern seas is modelled using shallow water equations. The inflow and outflow through the weather deck openings and over the bulwarks are modelled using boundary conditions in the flow domain. The hyperbolic flow equations are numerically solved using a higher-order finite volume method. The prescribed motion data for the weather deck is obtained directly from systematic model experiments. Simulated results of the deck water motion are compared against monitored data from the experiments, recorded using a system of pressure sensors and cameras placed on the weather deck. Additionally, the moment caused by the motion of the trapped water and its variation with the ship speed are investigated.

A high order Cartesian grid, finite volume method for elliptic interface problems

We present a higher-order finite volume method for solving elliptic PDEs with jump conditions on interfaces embedded in a 2D Cartesian grid. Second, fourth, and sixth order accuracy is demonstrated on a variety of tests including problems with high-contrast and spatially varying coefficients, large discontinuities in the source term, and complex interface geometries. We include a generalized truncation error analysis based on cell-centered Taylor series expansions, which then define stencils in terms of local discrete solution data and geometric information. In the process, we develop a simple method based on Green's theorem for computing exact geometric moments directly from an implicit function definition of the embedded interface. This approach produces stencils with a simple bilinear representation, where spatially-varying coefficients and jump conditions can be easily included and finite volume conservation can be enforced.

Three-dimensional high-order finite-volume method based on compact WENO reconstruction with hybrid unstructured grids

In the present work, a three-dimensional (3D) high-order finite-volume method, based on the compact simple weighted essentially non-oscillatory (WENO) reconstruction for hybrid unstructured grids, is developed. For each control cell, its stencils only involve neighboring cells. By ensuring that the various orders of the derivatives of the conservative variables are conserved over the stencils, an over-determined system of equations is formed, and the polynomials corresponding to each stencil are obtained by solving this system of equations using the least-squares method. The artificially determined weight coefficients, and the smoothing factors based on the smoothness of the flow field, are used to determine the nonlinear weights, which nonlinearly combine the polynomials of each stencil to obtain the final high-order reconstruction polynomial. Consequently, the conservative variables on the interface can be interpolated and the sphere-function based discrete gas-kinetic scheme (DGKS) is used to locally solve the Boltzmann equation. Lastly, the fluxes are calculated using the mesoscopic perspective based on the moment relations. The present algorithm has the following advantages: the stencils only involve the neighboring cells of each control volume, making the algorithm compact and simpler, especially for the operation of the boundary cells; the unknown coefficients are solved by the least-squares method to avoid the coefficient matrix being ill-conditioned; the simple WENO scheme artificially chooses the value to guarantee positive weights, in addition, the algorithm can be applied to 3D hybrid unstructured grids, making it more adaptable. In terms of flux calculation, DGKS not only inherits the advantages of the traditional gas-kinetic schemes (GKS) but also greatly simplifies the flux calculation expressions. Several cases are simulated to verify the correctness and robustness of the algorithm, and the accuracy test shows the algorithm is third-order accurate.

High-order implicit RBF-based differential quadrature-finite volume method on unstructured grids: Application to inviscid and viscous compressible flows

This paper exploits the potential of a high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method for effective simulation of inviscid and viscous compressible flows using unstructured grids. The framework of IRBFDQ-FV method is based on finite volume discretization which can guarantee conservation of mass, momentum and energy. The flow field variables within each control cell are represented by a fourth-order approximation constructed by a Taylor series expansion to the cell center with spatial derivatives as the undetermined coefficients. All spatial derivatives are computed by the meshless and highly converged RBF-based differential quadrature (RBFDQ) technique. Regarding flux evaluation, the discrete gas-kinetic flux solver is applied to evaluate inviscid and viscous fluxes simultaneously for compressible viscous flows. Resultantly, the special mathematical treatment for the viscous flux, which is usually adopted in other high-order methods, can be avoided. A point extrema-based extended bounds (PEEB) shock-capturing limiter is introduced to eliminate spurious numerical oscillations near discontinuities. Due to the implicit nature of the proposed method, implicit time-stepping schemes compatible with the spatial accuracy are devised to efficiently solve the resulting discretized equations. Representative compressible flow problems are simulated to verify the high-order accuracy, excellent efficiency and robustness of the present method. Comparison with two other high-order finite volume methods further shows competitiveness of the present method in solving compressible flow problems on unstructured grids.

Efficient numerical schemes for multidimensional population balance models

Multidimensional population balance models (PBMs) describe chemical and biological processes having a distribution over two or more intrinsic properties (such as size and age, or two independent spatial variables). The incorporation of additional intrinsic variables into a PBM improves its descriptive capability and can be necessary to capture specific features of interest. As most PBMs of interest cannot be solved analytically, computationally expensive high-order finite difference or finite volume methods are frequently used to obtain an accurate numerical solution. We propose a finite difference scheme based on operator splitting and solving each sub-problem at the limit of numerical stability that achieves a discretization error that is zero for certain classes of PBMs and low enough to be acceptable for other classes. In conjunction to employing specially constructed meshes and variable transformations, the scheme exploits the commutative property of the differential operators present in many classes of PBMs. The scheme has very low computational cost — potentially as low as just memory reallocation. Multiple case studies demonstrate the performance of the proposed scheme.

ENO-ET: a reconstruction scheme based on extended ENO stencil and truncated highest-order term

High-order numerical methods for hyperbolic balance laws must circumvent Godunov’s theorem, for which non-linear spatial reconstruction is a successful procedure; this allows the design of non-linear semi-discrete and fully discrete high-order finite volume methods. The Essentially-Non-Oscillatory (ENO) method pioneered by Harten and collaborators in 1987 was a fundamental breakthrough in providing the bases for constructing schemes of both high-order of accuracy in smooth regions and essentially non-oscillatory at discontinuities. Several variations of ENO have since appeared in the literature, notably the Weighted Essentially-Non-Oscillatory (WENO) method of Jiang and Shu. In the present article, after revisiting the main elements of reconstruction methods, including their analysis, we propose a new, conservative ENO-type reconstruction procedure, called ENO-ET, that preserves the main advantages of the classical ENO and successfully overcomes its well-known weaknesses. The newly proposed reconstruction scheme is systematically assessed and compared with the best existing reconstruction schemes. We do so in two ways: (i) the methods are purely regarded as local, non-linear interpolation objects and (ii) the reconstruction methods are implemented in the context of high-order fully-discrete ADER numerical schemes for solving hyperbolic balance laws. The resulting schemes are of arbitrary order of accuracy in both space and time. In this paper the methods are implemented up to seventh order of accuracy and tested on carefully chosen numerical examples. Tests solved include the linear advection equation, the Euler equations of gas dynamics and the blood flow equations. Particular features of interest here are: (a) attaining theoretically expected convergence rates for smooth solutions arising from demanding initial conditions, (b) compute essentially non-oscillatory profiles for discontinuous solutions and (c) attain successfully steady-steady state solutions by time marching. As compared to existing methods, the newly proposed ENO-ET scheme performs very well for all these problems.

A relaxed a posteriori MOOD algorithm for multicomponent compressible flows using high-order finite-volume methods on unstructured meshes

In this paper the relaxed, high-order, Multidimensional Optimal Order Detection (MOOD) framework is extended to the simulation of compressible multicomponent flows on unstructured meshes. The diffuse interface methods (DIM) paradigm is used that employs a five-equation model. The implementation is performed in the open-source high-order unstructured compressible flow solver UCNS3D. The high-order CWENO spatial discretisation is selected due to its reduced computational footprint and improved non-oscillatory behaviour compared to the original WENO variant. Fortifying the CWENO method with the relaxed MOOD technique has been necessary to further improve the robustness of the CWENO method. A series of challenging 2-D and 3-D compressible multicomponent flow problems have been investigated, such as the interaction of a shock with a helium bubble, and a water droplet, and the shock-induced collapse of 2-D and 3-D bubbles arrays. Such problems are generally very stiff due to the strong gradients present, and it has been possible to tackle them using the extended MOOD-CWENO numerical framework.

GP-MOOD: A positivity-preserving high-order finite volume method for hyperbolic conservation laws

We present an a posteriori shock-capturing finite volume method algorithm called GP-MOOD that solves a compressible hyperbolic conservative system at high-order solution accuracy (e.g., third-, fifth-, and seventh-order) in multiple spatial dimensions. The GP-MOOD method combines two methodologies, the polynomial-free spatial reconstruction methods of GP (Gaussian Process) and the a posteriori detection algorithms of MOOD (Multidimensional Optimal Order Detection). The spatial approximation of our GP-MOOD method uses GP's unlimited spatial reconstruction that builds upon our previous studies on GP reported in Reyes et al. (2018) [20] and Reyes et al. (2019) [21]. This paper focuses on extending GP's flexible variability of spatial accuracy to an a posteriori detection formalism based on the MOOD approach. We show that GP's polynomial-free reconstruction provides a seamless pathway to the MOOD's order cascading formalism by utilizing GP's novel property of variable (2R+1)th-order spatial accuracy on a multidimensional GP stencil defined by the GP radius R, whose size is smaller than that of the standard polynomial MOOD methods. The resulting GP-MOOD method is a positivity-preserving method. We examine the numerical stability and accuracy of GP-MOOD on smooth and discontinuous flows in multiple spatial dimensions without resorting to any conventional, computationally expensive a priori nonlinear limiting mechanism to maintain numerical stability.

A Very Easy High-Order Well-Balanced Reconstruction for Hyperbolic Systems with Source Terms

When adopting high-order finite volume schemes based on MUSCL reconstruction techniques to approximate the weak solutions of hyperbolic systems with source terms, the preservation of the steady states turns out to be very challenging. Indeed, the designed reconstruction must preserve the steady states under consideration in order to get the required well-balancedness property. A priori, to capture such a steady state, one needs to solve some strongly nonlinear equations. Here, we design a very easy correction to high-order finite volume methods. This correction can be applied to any scheme of order greater than or equal to 2, such as a MUSCL-type scheme, and ensures that this scheme exactly preserves the steady solutions. In contrast to usual techniques, our scheme avoids the inversion of the nonlinear function that governs the steady solutions. Moreover, for underdetermined steady solutions, several nonlinear functions must be considered simultaneously. Since the derived correction only considers the evaluation of the governing nonlinear functions, we are able to deal with underdetermined stationary systems. Several numerical experiments illustrate the relevance of the proposed well-balanced correction, as well as its main limitation, namely the fact that it may fail at being both well-balanced and more than second-order accurate for a specific class of initial conditions.

An implicit high-order radial basis function-based differential quadrature-finite volume method on unstructured grids to simulate incompressible flows with heat transfer

A high-order implicit radial basis function-based differential quadrature-finite volume (IRBFDQ-FV) method is presented in this work to efficiently simulate incompressible flows with heat transfer on unstructured mesh. The velocity and temperature fields are solved by locally using the lattice Boltzmann flux solver and the high-order finite volume method. Specifically, the proposed highly accurate finite volume method utilizes a high-order Taylor polynomial to approximate the solution within every control cell. Spatial derivatives are the corresponding coefficients in the polynomial, and they are approximated by the meshless radial basis function-based differential quadrature (RBFDQ) method. The diffusive and convective fluxes at each cell interface are simultaneously evaluated through local reconstruction of lattice Boltzmann solution using D2Q9 lattice velocity model. To efficiently calculate the solution with high-order accuracy, an implicit time-marching method incorporating the lower-upper symmetric Gauss-Seidel (LU-SGS) and the explicit first stage, singly-diagonally implicit Runge-Kutta (ESDIRK) approaches is devised. The proposed method is comprehensively validated by a series of numerical experiments containing both steady-state and time-dependent heat transfer problems with/without curved boundaries at a wide variety of Rayleigh numbers and Grashof numbers. The obtained results demonstrate a high degree of accuracy and reliability of the proposed method for complex flows on unstructured mesh. In comparison with the classical second-order method, the proposed high-order method has better computational efficiency when comparable results are achieved.

High-order finite volume method for linear elasticity on unstructured meshes

This paper presents a high-order finite volume method for solving linear elasticity problems on two-dimensional unstructured meshes. The method is designed to increase the effectiveness of finite volume methods in solving structural problems affected by shear locking.The particular feature of the proposed method is the use of Moving Least Squares (MLS) and Local Regression Estimators (LRE). Unlike other approaches proposed before, these interpolation schemes lead to a natural and simple extension of the classical finite volume method to arbitrary order. The unknowns of the problem are still the nodal values of the displacement which are obtained implicitly in a direct solution strategy.Some canonical tests are performed to demonstrate the accuracy of the method. An analytical example is considered to evaluate the sensitivity of the solution concerning the parameters of the algorithm. A thin curved beam and a crack problem are considered to show that the method can deal with the shear locking effect, stress concentrations, and geometries where unstructured meshes are required. An overall better behavior of the LRE is observed. A comparison between low and high-order schemes is presented, and a set of parameters for the interpolation method is found, delivering good results for the proposed cases.