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This paper is a rebuttal to the claim found in the literature that the MUSCL scheme cannot be third-order accurate for nonlinear conservation laws. We provide a rigorous proof for third-order accuracy of the MUSCL scheme based on a careful and detailed truncation error analysis. Throughout the analysis, the distinction between the cell average and the point value will be strictly made for the numerical solution as well as for the target operator. It is shown that the average of the solutions reconstructed at a face by Van Leer's kappa-scheme recovers a cubic solution exactly with kappa = 1/3, the same is true for the average of the nonlinear fluxes evaluated by the reconstructed solutions, and a dissipation term is already sufficiently small with a third-order truncation error. Finally, noting that the target spatial operator is a cell-averaged flux derivative, we prove that the leading truncation error of the MUSCL finite-volume scheme is third-order with kappa = 1/3. The importance of the diffusion scheme is also discussed: third-order accuracy will be lost when the third-order MUSLC scheme is used with a wrong fourth-order diffusion scheme for convection-diffusion problems. Third-order accuracy is verified by thorough numerical experiments for both steady and unsteady problems. This paper is intended to serve as a reference to clarify confusions about third-order accuracy of the MUSCL scheme, as a guide to correctly analyze and verify the MUSCL scheme for nonlinear equations, and eventually as the basis for clarifying third-order unstructured-grid schemes in a subsequent paper.

Sediment transport simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model coupling hydrodynamical and morphological processes was developed to simulate water flow, sediment transport, and morphological changes. Aiming at accurately predicting the sediment transport and sediment scouring processes, the model resolved the realistic features of sediment transport and used a GPU-based parallel computing technique to the accelerate calculation. This model was created in the framework of a Godunov-type finite volume scheme to solve the shallow water equations (SWEs). The SWEs were discretized into algebraic equations by the finite volume method. The fluxes of mass and momentum were computed by the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated by the proposed a splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which could run on GPUs to substantially accelerate the computation. The aim of the work was to develop a GPU-based numerical model to simulate hydrodynamical and morphological processes. The novelty is the application of the GPU techniques in the numerical model, making it possible to simulate the sediment transport and bed evolution in a high-resolution but efficient manner. The model was applied to two cases to evaluate bed evolution and the effects of the morphological changes on the flood patterns with high resolution. This indicated that the GPU-based high-resolution hydro-geomorphological model was capable of reproducing morphological processes. The computational times for this test case on the GPU and CPU were 298.1 and 4531.2 s, respectively, indicating that the GPU could accelerate the computation 15.2 times. Compared with the traditional CPU high-grid resolution, the proposed GPU-based high-resolution numerical model improved the reconstruction speed more than 2.0–12.83 times for different grid resolutions while remaining computationally efficient.

Debris flow simulations are important in practical engineering. In this study, a graphics processing unit (GPU)-based numerical model that couples hydrodynamic and morphological processes was developed to simulate debris flow, transport, and morphological changes. To accurately predict the debris flow sediment transport and sediment scouring processes, a GPU-based parallel computing technique was used to accelerate the calculation. This model was created in the framework of a Godunov-type finite volume scheme and discretized into algebraic equations by the finite volume method. The mass and momentum fluxes were computed using the Harten, Lax, and van Leer Contact (HLLC) approximate Riemann solver, and the friction source terms were calculated using the proposed splitting point-implicit method. These values were evaluated using a novel 2D edge-based MUSCL scheme. The code was programmed using C++ and CUDA, which can run on GPUs to substantially accelerate the computation. After verification, the model was applied to the simulation of the debris flow process of an idealized example. The results of the new scheme better reflect the characteristics of the discontinuity of its movement and the actual law of the evolution of erosion and deposition over time. The research results provide guidance and a reference for the in-depth study of debris flow processes and disaster prevention and mitigation.

In this paper, we present a finite-volume direct kinetic method, the so-called discrete unified gas kinetic scheme (DUGKS), for electrostatic plasma. One key feature of this method is the semi-implicit unsplitting treatment of particle transport and collision, and thus the time step in current DUGKS is not limited by the particle collision time. In addition, a fourth-order compact MUSCL scheme with a positivity preserving limiter is implemented in the interface reconstruction, which enables present DUGKS to preserve the favorable conservative property and positivity of distribution function. Combined with this MUSCL method, the semi-Lagrangian scheme is used for the particle transport in velocity space to remove Courant-Friedricks-Lewy restriction induced by the large electric force. As a result, the proposed DUGKS becomes an efficient and stable multiscale scheme. Several numerical experiments, including plasma sheath, linear Landau damping, collisional nonlinear Landau damping, and plasma ion acceleration, are performed to validate current DUGKS. A comparative study of current DUGKS with a general particle in cell (PIC) method which could handle particle collision in a conservative way is also presented. Theory and numerical experiments demonstrate that DUGKS is preferred for plasma flows involving small electrostatic perturbation and high collision regimes, while the PIC method is desired for the field- dominated plasma flows involving a wide range of velocities.

In this study, we propose a meshless scheme, GC-LSM (Geometric Conservation Least Squares Method), satisfying the geometric conservation and 1st order consistency. These constraints are introduced in order to overcome the non-conservativeness of the original meshless scheme and imposed by Lagrange multiplier on the least squares method which determines weighting coefficients of the derivative terms. Improvements on the meshless scheme are confirmed through computations with randomly distributed grid points for a sine wave, nozzle flow, and hypersonic flow around blunt body. Combined with AUSMPW + and MUSCL scheme, GC-LSM of the second order accuracy gives non-oscillating solution around a strong shockwave, even for hypersonic flow, and shows its capability comparable to the finite volume method in views of accuracy, robustness, and convergence.

A new hybrid optimized low-dissipation and adaptive MUSCL scheme is present for finite volume method. The proposed scheme, based on an optimized linear scheme with monotonicity preserving limitation and an adaptive MUSCL scheme, emphasizes on the resolution rather than the formal order. This technique is applied to cell interface reconstruction and bears similarity in form to the widely used MUSCL scheme except wider interpolation stencil. Although the scheme is not high order of accuracy because of adaptive MUSCL scheme acting as nonlinear part, the low-dissipation feature from the optimized linear part makes it very accurate and robust in many practical applications, where much richer flow structures can be obtained. A number of test cases are solved to validate the high resolution of the present scheme.

Author(s): Zhao, J; Ozgen, I; Liang, D; Hinkelmann, R | Abstract: In shallow water flow and transport modeling, the monotonic upstream-centered scheme for conservation laws (MUSCL) is widely used to extend the original Godunov scheme to second-order accuracy. The most important step in MUSCL-type schemes is MUSCL reconstruction, which calculate-extrapolates the values of independent variables from the cell center to the edge. The monotonicity of the scheme is preserved with the help of slope limiters that prevent the occurrence of new extrema during reconstruction. On structured grids, the calculation of the slope is straightforward and usually based on a 2-point stencil that uses the cell centers of the neighbor cell and the so-called far-neighbor cell of the edge under consideration. On unstructured grids, the correct choice for the upwind slope becomes nontrivial. In this work, 2 novel total variation diminishing schemes are developed based on different techniques for calculating the upwind slope and the downwind slope. An additional treatment that stabilizes the scheme is discussed. The proposed techniques are compared to 2 existing MUSCL reconstruction techniques, and a detailed discussion of the results is given. It is shown that the proposed MUSCL reconstruction schemes obtain more accurate results with less numerical diffusion and higher efficiency.

The numerical flux is an important element in the numerical model for solving shallow water flow problems in MUSCL scheme. It is used to determine the normal transport through the boundary of the control body. With the TVD numerical flux, because of the simple form of Lax–Friedrichs, it is adopted in many articles. In fact, there are many different numerical fluxes based on exact Riemann solutions or approximate Riemann solutions, and these numerical fluxes can be combined with the MUSCL scheme. For the one-dimensional and two-dimensional shallow water equations, starting from the accuracy and discontinuous capture ability, a series of numerical examples are simulated based on the six different numerical fluxes with MUSCL scheme. Through the result analysis, the paper find out the numerical fluxes which are more suitable for shallow water flow combined with MUSCL scheme.

In this article, we use a relaxation scheme for conservation laws to study liquid-vapor phase transition modeled by the van der Waals equation, which introduces a small parameter $e$ and a new variable. We solve the relaxation system in Lagrangian coordinates for one dimension and solve the system in Eulerian coordinates for two dimension. A second order TVD Runge-Kutta splitting scheme is used in time discretization and upwind or MUSCL scheme is used in space discretization. The long time behavior of the fluid is numerically investigated. If the initial data belongs to elliptic region, the solution converges to two Maxwell states. When the initial data lies in metastable region, the solution either remains in the same phase or converges to the Maxwell states depending to the initial perturbation. If the initial state is in the stable region, the solution remains in that region for all time.

To improve the modeling quality of pollutant transport in shallow waters, different reconstruction schemes have been proposed to better link the edge values to the centroid values of a pollutant concentration in finite-volume shallow water models: a scheme of higher (lower) order generally has a better (poorer) quantitative accuracy but lower (higher) computational efficiency. Here, a numerical comparative study of several classical schemes is first conducted under a variety of pollutant distribution conditions. The results reveal that, for the condition of relatively uniform pollutant distribution, the numerical accuracy of a lower-order scheme (such as the first-order scheme or the MUSCL scheme) may be similar to that of a higher-order scheme (such as the WENO scheme). The second-order derivative of the concentration, here termed the nonlinear indicator (NI), correlates well with the discrepancies between the numerical solutions and analytical solutions. A threshold value of approximately 10 − 7 ∼ 10 − 6 m-2 for the NI is identified, above which a higher-order scheme may be required. Based on this understanding, a hybrid first-order and WENO scheme is proposed. Numerical case studies show that the hybrid scheme can successfully combine the efficiency of the first-order scheme with the high accuracy of the WENO scheme for pollutant modeling.

In this paper, we introduce a mass conservative scheme for solving the Vlasov–Poisson equation. This scheme is based on an Eulerian approach and is constructed using an interpolation scheme with limiters. In order to preserve the mass, the difference in the values for numerical flux functions on each cell is used; for this, the flux functions are constructed by preserving both the solution along a characteristics and the mass in each cell. We mainly investigate the conservation of L 1 and L 2 norms of the distribution function, total energy, entropy, and minimum value. In addition, we show that this scheme is bounded on the total variation. To demonstrate the efficiency of the proposed scheme, this scheme is compared with the flux balance scheme, Positive and Flux Conservative scheme, Umeda’s scheme, and fifth order WENO reconstruction finite volume scheme.

The use of micro shock tubes has become common in many instruments requiring a high velocity and temperature flow field, for example in micro-propulsion systems and drug delivery devices for medical systems. A shock tube has closed ends, and the flow is generated by the rupture of a diaphragm separating a driver gas at high pressure from a driven gas at relatively low pressure. The rupture results in the movement of a shock wave and contact discontinuity into the low-pressure gas, and an expansion wave into the high pressure gas. The characteristics of the resulting unsteady flow for micro shock tubes are not well known as the physics of such tubes includes additional phenomena such as rarefaction and complex viscous effects at low Reynolds numbers. In the present study, computational fluid dynamics (CFD) calculations are made for unsteady compressible flow within a micro shock tube using the van-Leer MUSCL scheme and the two-layer k-e turbulence model. Novel results have been obtained and discussed of the effects of using different diaphragm pressure ratios, shock tube diameters and wall boundary conditions, namely no slip and slip walls.

A robust high-order scheme for the multi-phase flow computations featuring jumps and discontinuities due to shock waves and phase interfaces is presented. The scheme is based on high-order weighted-essentially non-oscillatory (WENO) finite volume schemes and high-order limiters to ensure the maximum principle or positivity of the various field variables including the density, pressure, and order parameters identifying each phase. The two-phase flow model considered besides the Euler equations of gas dynamics consists of advection of two parameters of the stiffened-gas equation of states, characterizing each phase. The design of the high-order limiter is guided by the findings of Zhang and Shu (2011) [36], and is based on limiting the quadrature values of the density, pressure and order parameters reconstructed using a high-order WENO scheme. The proof of positivity-preserving and accuracy is given, and the convergence and the robustness of the scheme are illustrated using the smooth isentropic vortex problem with very small density and pressure. The effectiveness and robustness of the scheme in computing the challenging problem of shock wave interaction with a cluster of tightly packed air or helium bubbles placed in a body of liquid water is also demonstrated. The superior performance of the high-order schemes over the first-order Lax-Friedrichs scheme for computations of shock-bubble interaction is also shown. The scheme is implemented in two-dimensional space on parallel computers using message passing interface (MPI). The proposed scheme with limiter features approximately 50% higher number of inter-processor message communications compared to the corresponding scheme without limiter, but with only 10% higher total CPU time. The scheme is provably second-order accurate in regions requiring positivity enforcement and higher order in the rest of domain.

形態: カラー図版あり, Physical characteristics: Original contains color illustrations, Accepted: 2017-04-01, 資料番号: PA1820013000

A new numerical scheme based on the finite-element method with a total-variation-diminishing property is developed with the aim of studying the shock-capturing capability of the combination. The proposed model is formulated within the framework of the one-step Taylor–Galerkin scheme in conjunction with the Van Leer’s limiter function. The approach is applied to the two-dimensional shallow water equations by different test cases, i.e., the partial, circular, and one-dimensional dam-break flow problems. For the one-dimensional case, the sub- and supercritical flow regimes are considered. The results are compared with the analytical, finite-difference, and smoothed particle hydrodynamics solutions in the literature. The findings show that the proposed model can effectively mask the sources of errors in the abrupt changes of the flow conditions and is able to resolve the shock and rarefaction waves where other numerical models produce spurious oscillations.

An idea of designing oscillation-less and high-resolution hybrid schemes is proposed and several types of hybrid schemes based on this idea are presented on block-structured grids. The general framework, for designing various types of hybrid schemes, is established using a Multi-dimensional Optimal Order Detection (MOOD) method proposed by Clain, Diot and Loubère [1]. The methodology utilizes low dissipation or dispersion but less robust schemes to update the solution and then implements robust and high resolution schemes to deal with problematic situations. A wide range of computational methods including central scheme, MUSCL scheme, linear upwind scheme and Weighted Essentially Non Oscillatory (WENO) scheme have been applied in the current hybrid schemes framework. Detailed numerical studies on classical test cases for the Euler system are performed, addressing the issues of the resolution and non-oscillatory property around the discontinuities.

A well-balanced van Leer-type numerical scheme for the shallow water equations with variable topography is presented. The model involves a nonconservative term, which often makes standard schemes difficult to approximate solutions in certain regions. The construction of our scheme is based on exact solutions in computational form of local Riemann problems. Numerical tests are conducted, where comparisons between this van Leer-type scheme and a Godunov-type scheme are provided. Data for the tests are taken in both the subcritical region as well as supercritical region. Especially, tests for resonant cases where the exact solutions contain coinciding waves are also investigated. All numerical tests show that each of these two methods can give a good accuracy, while the van Leer -type scheme gives a better accuracy than the Godunov-type scheme. Furthermore, it is shown that the van Leer-type scheme is also well-balanced in the sense that it can capture exactly stationary contact discontinuity waves.