Showing papers on "MUSCL scheme published in 2015"

An efficient unstructured MUSCL scheme for solving the 2D shallow water equations

Abstract: The aim of this paper is to present a novel monotone upstream scheme for conservation law (MUSCL) on unstructured grids. The novel edge-based MUSCL scheme is devised to construct the required values at the midpoint of cell edges in a more straightforward and effective way compared to other conventional approaches, by making better use of the geometrical property of the triangular grids. The scheme is incorporated into a two-dimensional (2D) cell-centered Godunov-type finite volume model as proposed in Hou et?al. (2013a,c) to solve the shallow water equations (SWEs). The MUSCL scheme renders the model to preserve the well-balanced property and achieve high accuracy and efficiency for shallow flow simulations over uneven terrains. Furthermore, the scheme is directly applicable to all triangular grids. Application to several numerical experiments verifies the efficiency and robustness of the current new MUSCL scheme. A novel MUSCL scheme is developed to provide more efficient and accurate numerical solutions to the 2D shallow water equations.The scheme is flexible for implementation on all common 2D unstructured grids.The scheme satisfies the well-balanced property.It can be generalized into 3D tetrahedron grids and to solve other hyperbolic PDEs.

High Resolution, Entropy-Consistent Scheme Using Flux Limiter for Hyperbolic Systems of Conservation Laws

Abstract: Existing entropy-consistent Euler flux avoids spurious oscillations and exactly preserves the stationary contact discontinuity but still leaves much room for further improvement in resolution and other applications. In this spirit, we propose a new high resolution entropy-consistent scheme to track discontinuities in hyperbolic systems of conservation laws. The new high resolution scheme, termed ECL/EC2L scheme, is based on two main ingredients: (1) the entropy-consistent flux, and (2) suitable flux limiter. And the same as entropy-consistent flux function, we obtain a high resolution entropy-consistent flux function (ECL-M-M/EC2L-M-M) precisely satisfies the discrete second law of thermodynamics. Several numerical simulations of the ECL/EC2L scheme have been tested on one-dimensional test cases. For Burgers equations, the ECL-M-M scheme exactly model rarefaction with a stationary shock and compression wave, and the numerical results are comparable to second order entropy consistent scheme. The second kind of case is the Euler equations with different initial value problems. The numerical results such as height, density, velocity and pressure are analyzed and then compared with the second order entropy-consistent scheme. The third kind of case is the shallow water equations with different kinds of dam break. Those excellent numerical results show the desired resolution and robustness of our ECL-M-M/EC2L-M-M scheme. Moreover, the ECL-M-M/EC2L-M-M flux is completely shock stable which will use to avoid multi-dimensional shock instability, particularly the carbuncle phenomenon.

Two-Dimensional Numerical Simulation of Gas–Solid Reactive Flow with Moving Boundary

Abstract: In order to combine the advantages of both the Lagrangian and Eulerian algorithm for a moving boundary, this work presents a two-dimensional axisymmetric computational model in ALE (arbitrary Lagrangian–Eulerian) forms for the gas–solid transient reacting flow with a moving boundary of the interior ballistic process. A two-phase flow model is established to describe the complex physical process based on a modified two-fluid theory, which takes into account gas production, interphase drag, intergranular stress, and heat transfer between two phases. The governing equations are discretized with the TVD-type MUSCL scheme to obtain a second-order accurate numerical method in finite volume form and solved by the semi-implicit method for pressure-linked equations with density corrections. A dynamic self-adapting mesh update method is developed to expand the computational domain for projectile motion and reduce the computational cost. Several verification tests demonstrate the accuracy and reliability of this app...

Solving the Savage–Hutter equations for granular avalanche flows with a second‐order Godunov type method on GPU

Abstract: Summary In this article, we apply Davis's second-order predictor-corrector Godunov type method to numerical solution of the Savage–Hutter equations for modeling granular avalanche flows. The method uses monotone upstream-centered schemes for conservation laws (MUSCL) reconstruction for conservative variables and Harten–Lax–van Leer contact (HLLC) scheme for numerical fluxes. Static resistance conditions and stopping criteria are incorporated into the algorithm. The computation is implemented on graphics processing unit (GPU) by using compute unified device architecture programming model. A practice of allocating memory for two-dimensional array in GPU is given and computational efficiency of two-dimensional memory allocation is compared with one-dimensional memory allocation. The effectiveness of the present simulation model is verified through several typical numerical examples. Numerical tests show that significant speedups of the GPU program over the CPU serial version can be obtained, and Davis's method in conjunction with MUSCL and HLLC schemes is accurate and robust for simulating granular avalanche flows with shock waves. As an application example, a case with a teardrop-shaped hydraulic jump in Johnson and Gray's granular jet experiment is reproduced by using specific friction coefficients given in the literature. Copyright © 2014 John Wiley & Sons, Ltd.

New Approaches to Limiting

Abstract: Of these, artificial viscosity was first employed by von Neumann and Richtmyer. It has been adopted in a sophisticated form by Jameson, but the general experience is that the adjustable parameters are not universal. We do not consider this possibility. Flux Corrected Transport was introduced by Boris and Book. The idea is to have available two different schemes, one that is accurate and another that is believed not to introduce spurious features, but is necessarily first-order and diffusive. For efficiency it should be very similar to the accurate scheme in its structure. If we were to take one step with the cautious scheme and then add back in the missing terms everywhere we would of course obtain the accurate result. Flux Corrected Transport adds back the missing terms in full only when it is safe to do so according to some criterion. Because they are added to the flux, they take from one cell and add to another in such a way as to sharpen existing gradients. Gradient limiting originated with the MUSCL scheme proposed by van Leer In a finite volume scheme the data consists solely of cell average values, and in order to obtain better than first-order results, more than this is needed. In the original version only a gradient was added, by constructing two more or less equally plausible candidate gradients and then taking a nonlinear average, such as the harmonic mean, that is biased toward the smaller alternative. van Leer gave an appealing graphical interpretation of this, allowing a visual proof that this procedure could not create overshoots. This results in a reconstruction that is generally discontinuous across cell boundaries. Most writers, including van Leer, have taken these discontinuities literally, introducing Riemann solvers to resolve them. This is not harmful in one dimension, because it simply results in waves that travel either left or right, not differently from other waves in the solution. However, in more than one dimension, taking the discontinuous reconstruction to be genuine, introduces strong one-dimensional waves that travel perpendicular to the arbitrary cell boundary, and may have no counterpoint in the actual flow. Various authors have generalized this procedure to include higher-order reconstructions, in which the higher derivatives are also evaluated in a way that is biased toward caution. A further classification of limiting turns on how much information is available about constraints on the solution. For example, neither density nor pressure may be negative, and speeds may not exceed the speed of light. In one-dimensional hyperbolic systems, if they are linear, the Riemann invariants may not take on new values. In pure advection problems, the value at (x, t) is bounded by the values along the particle path

Higher-order MUSCL scheme for transport equation originating in a neuronal model

Abstract: We consider a transport equation with mixed boundary conditions (Dirichlet and Neumann) originating in a neuronal model. This equation contains point-wise delay and advanced argument. The objective of this paper is to construct and analyze higher order MUSCL scheme to find a numerical solution. The developed scheme is stable and convergent. The importance of this scheme is that it is valid for large as well as for small values of point-wise delay and advance. Some test examples are included to examine the behavior of the solution and to verify the order of convergence. The effect of point-wise delay and advance arguments on the solution is shown graphically.

A scheme for inviscid compressible flow, considering a gas in thermo-chemical equilibrium

Abstract: A finite volume scheme for the solution of the unsteady 1D Euler equations, considering the working gas in thermo-chemical equilibrium, is presented. To achieve total variation diminishing (TVD) properties in the numerical scheme, a technique proposed by the co-authors, based on the use of different limiter functions in each wave of the Riemann problem, is applied. By proper selection of the limiter functions, the unwanted effects of the numerical viscosity on the capture of contact discontinuities are reduced, but without losing robustness in shock waves resolution. With the aim of evaluating the developed numerical scheme, results obtained solving several Riemann problems, specially selected for this specific purpose, are presented.

A Finite Volume Scheme for Three-Dimensional Diffusion Equations

Abstract: The extension of diamond scheme for diffusion equation to three dimensions is presented. The discrete normal flux is constructed by a linear combination of the directional flux along the line connecting cell-centers and the tangent flux along the cell-faces. In addition, it treats material discontinuities by a new iterative method. The stability and first-order convergence of the method is proved on distorted meshes. The numerical results illustrate that the method appears to be approximate second-order accuracy for solution. AMS subject classifications: 65M08, 65M12, 65M55

Near Field Sonic Boom Calculation of Benchmark Cases

Abstract: This paper validates the accuracy of a high order accuracy CFD code on predicting near field signatures of sonic boom propagation. The flow near the body is solved using a structured grid Euler solver with low diffusion E-CUSP schemes, 3rd order MUSCL scheme, and 3rd and 5th order WENO schemes. Three benchmark cases, including two non-lifting models and one lifting model, are calculated. The predicted results are in good agreement with that of experiment data. For the Delta wing configuration, the mesh provided by the workshop generates a smooth expansion wave at the off-track plane. When the mesh is refined, a shock coalesced by the compression waves from the wing tip is captured and it generates a weak shock wave interacting with the expansion.

Design of h-Darrieus vertical axis wind turbine

Abstract: Numerical simulation is used to predict the performance of a Vertical Axis Wind Turbine (VAWT) H-Darrieus. The rotor consists of three straight blades with shape of aerofoil of the NACA family attached to a rotating vertical shaft. The influence of the solidity is tested to get design tendencies. The mesh has two fluid volumes: one sliding mesh for the rotor where the rotation velocity is established while the other is the environment of the rotor. Bearing in mind the overall flow is characterized by important secondary flows, the turbulence model selected was realizable k-epsilon with non-equilibrium wall functions. Conservation equations were solved with a Third-Order Muscl scheme using SIMPLE to couple pressure and velocity. During VAWT operation, the performance depends mainly on the relative motion of the rotating blade and has a fundamental period which depends both on the rate of rotation and the number of blades. The transient study is necessary to characterise the hysteresis phenomenon. Hence, more than six revolutions get the periodic behaviour. Instantaneous flows provide insight about wake structure interaction. Time averaged parameters let obtain the characteristic curves of power coefficient.

High Order Evaluation of S-76 in Hover

Abstract: The aerodynamics characteristics of the S-76 rotor in hover have been studied on stretched non-orthogonal grids using spatially high order symmetric total variation diminishing (STVD) schemes. Several companion numerical viscosity terms have been tested. A baseline finite volume methodology termed TURNS (Transonic Unsteady Rotor Navier-Stokes) is the starting point for this effort. The original TURNS solver solves the 3-D compressible Navier-Stokes equations in an integral form using a third order upwind scheme. It is first or second order accurate in time. In the modified solver, the inviscid flux at a cell face is decomposed into two parts. The first part of the flux is symmetric in space, while the second part consists of an upwind-biased numerical viscosity term. The symmetric part of the flux at the cell face is computed to fourth-, sixthor eighth order accuracy in space. The numerical viscosity portion of the flux is computed using either a third order accurate MUSCL scheme or a fifth order WENO scheme. A number of results are presented for the S-76 rotor in hover. Comparisons with the baseline third order upwind scheme and experiments are given.

Development of High-Resolution Total Variation Diminishing Scheme for Linear Hyperbolic Problems

Abstract: A high-resolution, total variation diminishing (TVD) stable scheme is derived for scalar hyperbolic problems using the method of flux limiters. The scheme was constructed by combining the 1st-order upwind scheme and the 3rd-order quadratic upstream interpolation scheme (QUICK) using new flux limiter function. The new flux limiter function was established by imposing several conditions to ensure the TVD properties of the scheme. For temporal discretization, the theta method was used, and values for the parameter θ were chosen such that the scheme is unconditionally stable. Numerical results are presented for one-dimensional pure advection problems with smooth and discontinuous initial conditions and are compared to those of other known numerical schemes. The results show that the proposed numerical method is stable and of higher order than other common schemes.

Total variation diminishing finite volume schemes for one dimensional advection-diffusion equation and the relationship between flux limiter and mesh parameters

Abstract: Finite volume schemes for one dimensional Advection-Diffusion Equation (ADE) are discussed in this article. As a result, a general explicit difference equation of the form Un+1 m = aU n m−1+ bU n m+ cU n m+1 is obtained with general coefficients a, b, and c. Stability condition and local truncation error for this general form of explicit difference equation are derived. Then, total Variation Diminishing (TVD) schemes for general flux limiter ψ(r) are also discussed. Further, a relation between flux limiter and mesh length parameters is also obtained. Numerical justification for order of convergence for upwind, central difference and various TVD schemes are also presented. AMS Subject Classification: 65M08, 65M12, 65M15, 65N08, 65N12

Influence of Solidity on Vertical Axis Wind Turbines

Abstract: This paper aims to predict the performance of a VAWT. The H-type Darrieus turbine consists of three straight blades with shape of aerofoil attached to a rotating vertical shaft. Bearing in mind the overall flow is characterized by important secondary flows, the turbulence model selected was realizable k-epsilon with non-equilibrium wall functions. Conservation equations were solved with a Third-Order MUSCL scheme using SIMPLE to couple continuity and momentum equations. A parametric study has been carried out to analyse the solidity effect on the non-dimensional curves so that the range of tip speed ratio of operation could be predicted as well as their self-starting behaviour.

MUSCL reconstruction and Haar wavelets

Abstract: MUSCL extensions (Monotone Upstream-centered Schemes for Conservation Laws) of the Godunov numerical scheme for scalar conservation laws are shown to admit a rather simple reformulation when recast in the formalism of the Haar multi-resolution analysis of L(R). By pursuing this wavelet reformulation, a seemingly new MUSCL-WB scheme is derived for advection-reaction equations which is stable for a Courant number up to 1 (instead of roughly 1 2 ). However these highorder reconstructions aren’t likely to improve the handling of delicate nonlinear wave interactions in the involved case of systems of Conservation/Balance laws.

A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations

Abstract: A total variation diminishing-weighted average flux (TVD-WAF)-based hybrid numerical scheme for the enhanced version of nonlinearly dispersive Boussinesq-type equations was developed. The one-dimensional governing equations were rewritten in the conservative form and then discretized on a uniform grid. The finite volume method was used to discretize the flux term while the remaining terms were approximated with the finite difference method. The second-order TVD-WAF method was employed in conjunction with the Harten-Lax-van Leer (HLL) Riemann solver to calculate the numerical flux, and the variables at the cell interface for the local Riemann problem were reconstructed via the fourth-order monotone upstream-centered scheme for conservation laws (MUSCL). The time marching scheme based on the third-order TVD Runge-Kutta method was used to obtain numerical solutions. The model was validated through a series of numerical tests, in which wave breaking and a moving shoreline were treated. The good agreement between the computed results, documented analytical solutions, and experimental data demonstrates the correct discretization of the governing equations and high accuracy of the proposed scheme, and also conforms the advantages of the proposed shock-capturing scheme for the enhanced version of the Boussinesq model, including the convenience in the treatment of wave breaking and moving shorelines and without the need for a numerical filter.

Separated Flow Prediction Around a 6:1 Prolate Spheroid Using Reynolds Stress Models

Abstract: A numerical study of the separated flow about a 6:1 prolate spheroid at high-angle of attack using state-of-the-art Reynolds stress models is presented. The convective fluxes of the mean-flow and the Reynolds stress model equations are approximated by a third-order upwind biased MUSCL scheme. The diffusive flux is approximated by second-order central differencing based on a full-viscous stencil. The objective is to evaluate the applicability of RSM to realistic high-Reynolds separated flows. Comprehensive comparisons of the boundary layer velocity profile and of the Reynolds stress tensor components against the experimental data are presented. A very good agreement between the experimental measurements and calculated boundary layer velocity profiles is obtained. However, only reasonable agreement is obtained for the Reynolds stress components. It is shown that the common first-order upwind approximation of the Reynolds stress model convective flux alone may adversely affect the accuracy of the solution.

The research on the hydraulic numerical simulation of flux limiter function

Abstract: The advantages of the form of this paper is numerical calculation without spurious numeric, and keeps monotonicity of numerical value. It contrasts the difference among various flux limiter flux limiter function in solving equations by examples and verifies the advantages which is conservative, robustness, spurious numericalosc and high-resolution catches discontinuity when introducing TVD flux limiter function Finite Volume Method model in numerical simulation. 1.Introduction The advantages of TVD form is great keeping the monotonicity of numerical solution[1], efficiently weakening numerical osc which produces in cell boundaries, having high solution to discontinuity. But its disadvantage is the precision of TVD form can reach second order at most, and it only has first order at local extremum, which makes extreme value attenuation. And there isn't super-first order TVD form on multidimensional problems[2]. This paper is based on Roe’s approximate Riemann solution[3], gets high-order accuracy shallow water equations discrete form by reconstitution and flux limiter function, and compares the merits and demerits in solving equations of flux limiter function in solving equations by examples. 2. The discretization of the governing equations 2.1 Governing equation and numerical discretization The conservation form of vertically averaged two-dimension governing equation, U f g S t x y ∂ ∂ ∂ + + = ∂ ∂ ∂ (1)

Numerical Solution of Hyperbolic Problems

Abstract: In this chapter, we introduce spatial discretization schemes for systems of conservation laws. For smooth problems, summation-by-parts operators with weak enforcement of boundary conditions allow for the design of stable high-order accurate schemes. Summation by parts is the discrete equivalent of integration by parts and the matrix operators that are presented lead to energy estimates that in turn lead to provable stability. The discrete stability analysis follows naturally from the continuous analysis of well-posedness. For non-smooth problems, the need to accurately capture multiple solution discontinuities of hyperbolic stochastic Galerkin systems requires the introduction of shock-capturing methods. In this setting, we outline the MUSCL scheme with flux limiters and the HLL Riemann solver. We also briefly discuss how to add artificial dissipation and an issue regarding time integration.