Shock-capturing Ability Research Articles

A staggered Lagrangian magnetohydrodynamics method based on subcell Riemann solver

This paper uses a general formalism to derive staggered Lagrangian method for 2D compressible magnetohydrodynamics (MHD) flows. A subcell method is introduced to discretize the MHD system and some Riemann problems over subcells are solved at the cell center and grid node respectively. In these solvers, only the fast-waves in all jumping relations are considered and thus the solution structure is simple. The discrete conservations of mass, momentum and energy are preserved naturally in the proposed numerical method. In order to meet the thermodynamic Gibbs relation in isentropic flows, an adaptive Riemann solver is implemented at the cell center, in which a criterion is proposed to reduce overheating errors in the rarefying problems and maintains the excellent shock-capturing ability simultaneously. It is worth to be noticed that the divergence-free condition is naturally satisfied in the Lagrangian method. Various numerical tests are presented to demonstrate the accuracy and robustness of the algorithm.

High-order accurate well-balanced energy stable adaptive moving mesh finite difference schemes for the shallow water equations with non-flat bottom topography

This paper proposes high-order accurate well-balanced (WB) energy stable (ES) adaptive moving mesh finite difference schemes for the shallow water equations (SWEs) with non-flat bottom topography. To enable the construction of the ES schemes on moving meshes, a reformulation of the SWEs is introduced, with the bottom topography as an additional conservative variable that evolves in time. The corresponding energy inequality is derived based on a modified energy function, then the reformulated SWEs and energy inequality are transformed into curvilinear coordinates. A two-point energy conservative (EC) flux is constructed, and high-order EC schemes based on such a flux are proved to be WB that they preserve the lake at rest. Then high-order ES schemes are derived by adding suitable dissipation terms to the EC schemes, which are newly designed to maintain the WB and ES properties simultaneously. The adaptive moving mesh strategy is performed by iteratively solving the Euler-Lagrangian equations of a mesh adaptation functional. The fully-discrete schemes are obtained by using the explicit strong-stability preserving third-order Runge-Kutta method. Several numerical tests are conducted to validate the accuracy, WB and ES properties, shock-capturing ability, and high efficiency of the schemes.

Path-conservative positivity-preserving well-balanced finite volume WENO method for porous shallow water equations

The present study constructed a path-conservative high-order positivity-preserving well-balanced finite volume Riemann solver for the one-dimensional porous shallow water equations with discontinuous porosity and bottom topography. First, the finite difference equations were formulated using the path-conservative approach to treat non-conservative products in source terms. As the solution depends on the paths, the family of paths was formulated for the solutions to satisfy the mass and energy conservations over the smooth paths connecting the discontinuous geometry (porosity and/or bottom topography). Such a property was achieved using the stationary wave, a stationary weak solution of the porous shallow water equations. Second, to increase computational accuracy, the weighted essentially non-oscillatory (WENO) and the Runge-Kutta methods were implemented for the spatiotemporal discretization of the porous shallow water equations, respectively. Third, a positivity-preserving limiter was employed to provide robustness and efficiency in handling wet and dry problems. Finally, all procedures in the present model were formulated to ensure the well-balanced property (i.e., the exact conservation of initial still water conditions over the irregular geometry). A set of numerical experiments was performed to verify that the present model exhibits positivity-preserving and well-balanced properties along with high-order accuracy and the shock-capturing ability for all types of Riemann problems, including eight wet cases and five dry cases. The numerical results of all cases agree with the analytical solutions.

Improvements of the fifth-order WENO-JS-type scheme with normalized smoothing factor for gas dynamic Euler equations

In this paper, we developed a novel high-resolution fifth-order WENO-JS-type scheme that can achieve the optimal accuracy order at or near the arbitrary order critical points. To enhance the accuracy of the weight function of WENO-JS, we normalized the local smoothness factor to obtain a series of normalized variables. By introducing a mapping function that can increase the accuracy of the normalization variables, we developed a new normalized WENO-JS-type scheme called WENO-NSI. The WENO-NSI scheme can improve the shock capture ability by selecting the apposite parameter λ in the mapping function. Numerical experiments of one- and two-dimensional gas dynamic Euler equations for the present WENO-NSI scheme are compared with those of the WENO-JS/Z/M/D/IM and TENO-Z schemes. The results show that the present scheme can achieve theoretical accuracy at or near the arbitrary order critical points, and it has a better shock capturing ability than other schemes.

A multi-objective Bayesian optimization environment for systematic design of numerical schemes for compressible flow

Multi-objective Bayesian optimization (MOBO) is an efficient and robust optimization framework for expensive functions. In this work, we use MOBO to optimize the free parameters of a high-order nonlinear weighted essentially non-oscillatory (WENO) reconstruction scheme to devise a model for implicit large eddy simulations. We concurrently optimize for a low dispersion error and sufficient shock-capturing ability for compressible flows as well as for physically consistent transition occurring in under-resolved flow regions. With our approach, we follow the genealogy of designing implicit sub-grid models. Yet, in contrast to previous works that were limited to incompressible flows, our model is also applicable to compressible flows. Validated results show that the model is able to decrease excessive dissipation in continuous flow regimes, to capture shocks with little dispersive and dissipative errors while achieving a well shaped vortical structures. The proposed framework is general and can be used to design a physically consistent numerical scheme for under-resolved compressible-flow simulations.

Granular Flow–Obstacle Interaction and Granular Dam Break Using the S-H Model with the TVD-MacCormack Scheme

An accurate second-order spatial and temporal finite-difference scheme is applied to solve the dynamics model of a depth-averaged avalanche. Within the framework of the MacCormack scheme, a total variation diminishing term supplements the corrector step to suppress large oscillations in domains with steep gradients. The greatest strength of the scheme lies in its high computational efficiency while maintaining satisfactory accuracy. The performance of the scheme is tested on a granular flume flow–obstacle interaction scenario and a granular dam breaking scenario. In the former, the flume flow splits into two granular streams when an obstacle is encountered. The opening between the two granular streams widens when the side length of the obstacle increases. In the simulation, shock waves with a fan-shaped configuration are captured, and successive waves in the tail of the avalanche between the two streams are observed. In the latter scenario, the average values and the fluctuations in the flow rate and velocity (at relatively steady state) decrease with the width of the breach. The capture of complex and typical granular-flow phenomena indicates the applicability and effectiveness of combining the TVD-MacCormack Scheme and S-H model to simulate dam breaking and inclined flow–obstacle interaction cases. In this study, the dense granular flow strikes on a rigid obstacle that is described by a wall boundary, rather than a topographic feature with a finite slope. This shows that the TVD-MacCormack scheme has a shock-capturing ability. The results of granular dam break simulations also revealed that the boundary conditions (open or closed) affect the collapse of the granular pile, i.e., the grains evenly breached out under closed boundary conditions, whereas the granules breaching out of the opening were mostly grains adjacent to the boundaries under open boundary conditions.

High-order accurate entropy stable adaptive moving mesh finite difference schemes for special relativistic (magneto)hydrodynamics

This paper develops high-order accurate entropy stable (ES) adaptive moving mesh finite difference schemes for the two- and three-dimensional special relativistic hydrodynamic (RHD) and magnetohydrodynamic (RMHD) equations, which is the high-order accurate extension of Duan and Tang (2021) [23]. The key point is the derivation of the higher-order accurate entropy conservative (EC) and ES finite difference schemes in the curvilinear coordinates by carefully dealing with the discretization of the temporal and spatial metrics and the Jacobian of the coordinate transformation and constructing the high-order EC and ES fluxes with the discrete metrics. The spatial derivatives in the source terms of the symmetrizable RMHD equations and the geometric conservation laws are discretized by using the linear combinations of the corresponding second-order case to obtain high-order accuracy. Based on the proposed high-order accurate EC schemes and the high-order accurate dissipation terms built on the WENO reconstruction, the high-order accurate ES schemes are obtained for the RHD and RMHD equations in the curvilinear coordinates. The mesh iteration redistribution or adaptive moving mesh strategy is built on the minimization of the mesh adaption functional. Several numerical tests are conducted to validate the shock-capturing ability and high efficiency of our high-order accurate ES adaptive moving mesh methods on the parallel computer system with the MPI communication. The numerical results show that the high-order accurate ES adaptive moving mesh schemes outperform both their counterparts on the uniform mesh and the second-order ES adaptive moving mesh schemes.

A hybrid kinetic WGVC–WENO scheme for compressible flows

In this paper, we propose a hybrid kinetic weighted group velocity control–weighted essentially non-oscillatory (WGVC–WENO) scheme which combines the WGVC–WENO reconstruction technique He et al. (2014) with the hybrid kinetic method proposed in Liu et al. (2015). The new scheme is based on the idea of hybridization. The hybrid kinetic method is adopted for flux calculations, and the WGVC–WENO technique is adopted for physical reconstructions. In smooth flow regions, the collision-related flux method and WGVC reconstruction technique play a leading role. The kinetic flux vector splitting (KFVS) method and WENO reconstruction technique are dominant in discontinuous regions. The third-order TVD Runge–Kutta method is used for temporal discretization. Many one-dimensional and two-dimensional numerical examples show that the new scheme has the properties of high accuracy, low dissipation, and good shock-capturing ability.

High order semi-implicit weighted compact nonlinear scheme for viscous Burgers’ equations

This paper develops a high order semi-implicit weighted compact nonlinear scheme (WCNS) for one- and two-dimensional viscous Burgers’ equations. This semi-implicit WCNS combines a fifth-order WCNS with a third-order implicit-explicit Runge–Kutta (IMEX-RK) time-stepping method. The advection terms of viscous Burgers’ equations are treated explicitly, while the viscosity terms are treated implicitly. Stability analysis shows that the CFL condition of the semi-implicit WCNS is controlled only by the advection terms. Compared to the explicit time-stepping method, the semi-implicit method has the advantage in terms of computational efficiency. Numerical results validate the accuracy and efficiency of the semi-implicit WCNS. This method can also solve viscous Burgers’ equations with large Reynolds numbers and has high-resolution shock-capturing ability.

Assessment of Upwind/Symmetric WENO Schemes for Direct Numerical Simulation of Screech Tone in Supersonic Jet

Screech tones are the high intensity shock-associated noise with discrete frequency in imperfectly supersonic jet. Accuracte numerical simulation of shock-associated noise requires numerical scheme with high order accuracy, low dissipation and low dispersion as well as robust shock-capturing ability. The applicability of eight kinds of upwind/symmetric WENO schemes for the direct numerical simulation of screech tone is evaluated through comparison study on the spectral characteristics and direct numerical simulations of underexpanded supersonic cold jet issuing from circular sonic nozzle. The spectral characteristics based on the approximate dispersion relation shows that the adaptive central-upwind (WENO-SYMCU-6) scheme has the best resolution and the minimum value of point-per-wavelength. However, the comparison of nonlinear response indicates that the improved upwind WENO-ZM-5 ( $$p=1$$ ) scheme has almost the weakest nonlinear response for the single Fourier mode. And the performances in the direct numerical simulations also show that the WENO-ZM-5 ( $$p=1$$ ) scheme is the best. With this method, not only the screech tones of the high amplitude axisymmetric mode are obtained, but also a variety of high-frequency coupling modes with the highest amplitudes can be distinguished in a wider spectrum range. And there is no nonphysical high-frequency wave induced by the nonlinear implementation of WENO schemes at the near region of the nozzle exit lip.

High-Order Hybrid WCNS-CPR Scheme for Shock Capturing of Conservation Laws

By introducing hybrid technique into high-order CPR (correction procedure via reconstruction) scheme, a novel hybrid WCNS-CPR scheme is developed for efficient supersonic simulations. Firstly, a shock detector based on nonlinear weights is used to identify grid cells with high gradients or discontinuities throughout the whole flow field. Then, WCNS (weighted compact nonlinear scheme) is adopted to capture shocks in these areas, while the smooth area is calculated by CPR. A strategy to treat the interfaces of the two schemes is developed, which maintains high-order accuracy. Convergent order of accuracy and shock-capturing ability are tested in several numerical experiments; the results of which show that this hybrid scheme achieves expected high-order accuracy and high resolution, is robust in shock capturing, and has less computational cost compared to the WCNS.

An Efficient Low-Dissipation Hybrid Central/WENO Scheme for Compressible Flows

This paper develops a new hybrid finite difference scheme which consists of a nonlinear WENOCU4 scheme to maintain the numerical stability near discontinuities and a fourth-order linear central scheme elsewhere to accurately resolve smooth fluctuations and to speed up the computation. The central scheme is constructed in a robust skew-symmetric form which satisfies energy conservation property. A new efficient discontinuity detector is employed to identify the smoothness of the flowfield. As for the one-dimensional scalar equations, the hybrid scheme behaves almost no attenuation associated with propagation error. The quantitative analysis of the shock-capturing error is investigated. The hybrid scheme also exhibits high-resolution spectral property in the wavenumber space. Extensive cases of Euler equations further demonstrate the high-resolution shock-capturing ability of the detector, remarkable vortices-resolving capacity, and superior computational efficiency of the hybrid scheme. The reduced computational cost of the hybrid scheme is about 30−50% with respect to the baseline scheme.

Piston-Driven Numerical Wave Tank Based on WENO Solver of Well-Balanced Shallow Water Equations

A numerical wave tank equipped with a piston type wave-maker is presented for long-duration simulations of long waves in shallow water. Both wave maker and tank are modelled using the nonlinear shallow water equations, with motions of the numerical piston paddle accomplished via a linear mapping technique. Three approaches are used to increase computational efficiency and accuracy. First, the model satisfies the exact conservation property (C-property), a stepping stone towards properly balancing each term in the governing equation. Second, a high-order weighted essentially non-oscillatory (WENO) method is used to reduce accumulation of truncation error. Third, a cut-off algorithm is implemented to handle contaminated digits arising from round-off error. If not treated, such errors could prevent a numerical scheme from satisfying the exact C-property in long-duration simulations. Extensive numerical tests are performed to examine the well-balanced property, high order accuracy, and shock-capturing ability of the present scheme. Correct implementation of the wave paddle generator is verified by comparing numerical predictions against analytical solutions of sinusoidal, solitary, and cnoidal waves. In all cases, the model gives satisfactory results for small-amplitude, low frequency waves. Error analysis is used to investigate model limitations and derive a user criterion for long wave generation by the model.

An Adaptive High-Resolution and Low-Dissipation Hybrid Energy Consistent/WENOCU Scheme

We investigate the performance of the hybrid energy consistent/WENOCU4 scheme compared to the classical WENOCU4 scheme. It is borne out that the hybrid scheme preserves excellent shock-capturing ability without numerical oscillations. In the meantime, the hybrid scheme can achieve more accurate representations of smooth flow features which indicates its low-dissipation capacities. Moreover, hybrid scheme significantly outperforms WENOCU4 for its superior computational efficiency. Therefore, employing hybrid scheme is a much more cost-effective choice for the simulation of shock/turbulence interaction.

Characteristics of hydrogen jet combustion in a high-enthalpy supersonic crossflow

The facilitation of a stable combustion process is of utmost importance for the realizability and performance of hypersonic propulsion systems. To elucidate the turbulent combustion characteristics, wall-modeled large eddy simulations of a transverse jet injection into a heated supersonic flow are conducted employing a detailed reaction mechanism. The computation framework utilizes an adaptive central-upwind weighted essentially nonoscillatory (WENO-CU) scheme to achieve the sixth-order accuracy in smooth flowfields, while keeping a good shock-capturing ability. The reacting zones agree well with experimental measurements in terms of the instantaneous distribution of OH radicals. And the flame penetration height has been predicted with an error of less than 17%. It is found that the turbulent reacting flow is dominated by nonpremixed combustion mainly taking place in the near-wall region and jet windward shear-layer. Moreover, the autoignition process, which plays a critical role in stabilizing supersonic combustion, shows to favor a fuel-lean or not very fuel-rich environment of a high enthalpy. Local scalar dissipation induced by turbulence gives rise to a rapid fuel mixing with the surrounding air. However, this effect may also lead to the decrease in local temperature.

Numerical investigation of a sandbar formation and evolution in a tide-dominated estuary using a hydro-morphodynamic model

ABSTRACT This paper presents a numerical study on ‘sandbar’ feature in a tide-dominated symmetrical funnel-shaped estuary using a hydro-morphodynamic model. A series of benchmark flows is simulated to demonstrate the capabilities of the model (e.g., well-balanced property, treating wet/dry fronts, shock-capturing ability), and the good quantitative accuracy. Afterwards, the model is applied for systematic numerical case studies. The main geometrical parameters, the upstream and boundary conditions are specified with reference to the Qiantang River Estuary (QRE), where a giant ‘sandbar’ is well known. It is shown that a maximum turbidity zone forms due to interactions of the riverine discharge and the tidal flow. Strong deposition occurs in the maximum turbidity zone, whereas erosion occurs otherwise. This results in a sandbar of comparable magnitude and position with the observed ‘sandbar’ in the QRE. As the sandbar grows, a slight increasing trend followed by a rapid decreasing trend for the difference between high and low water levels is established. A larger riverine discharge leads to a sandbar more closer to the seaside, but effect of the seasonal discharge variation is limited. Erodibility of bed and seaside sediment input mainly influence the height of the formed sandbar.

Numerical dissipation control in an adaptive WCNS with a new smoothness indicator

A novel adaptive central-upwind weighted compact nonlinear scheme (WCNS) based on an adjusting parameter of the smoothness indicator is proposed. For the purposes of restraining numerical dissipation in smooth regions, preserving shock-capturing ability and avoiding spurious numerical oscillations around discontinuous regions, an adaptive parameter is introduced in this improved WCNS. The improved scheme can automatically modify the adaptive parameter to fit nonlinear weights to discontinuity according to the local flow-field properties obtained by a discontinuity detector. In smooth regions, the scheme inclines to an optimal central scheme to minimize dissipations and capture turbulent features. While, in discontinue regions, it is more likely to behave as an up-wind scheme for stable shock-capturing ability and numerical robustness. Furthermore, to solve the problem of losing accuracy, a smoothness indicator with a mapping function is introduced. The order-recover property of the improved scheme can obviously avoid unsymmetrical breaking of the flow field. A variety of benchmark cases are tested to verify the improved WCNS performance. The computational efficiencies of all cases have been estimated by measuring the computational time. Numerical results demonstrate that the improved WCNS exhibits excellent shock-capturing ability, lower numerical dissipation, higher numerical robustness and efficiency in resolving complex flow features.

A high-order positivity-preserving single-stage single-step method for the ideal magnetohydrodynamic equations

We propose a high-order finite difference weighted ENO (WENO) method for the ideal magnetohydrodynamics (MHD) equations. The proposed method is single-stage (i.e., it has no internal stages to store), single-step (i.e., it has no time history that needs to be stored), maintains a discrete divergence-free condition on the magnetic field, and has the capacity to preserve the positivity of the density and pressure. To accomplish this, we use a Taylor discretization of the Picard integral formulation (PIF) of the finite difference WENO method proposed in Christlieb et al. (2015) [23] , where the focus is on a high-order discretization of the fluxes (as opposed to the conserved variables). We use the version where fluxes are expanded to third-order accuracy in time, and for the fluid variables space is discretized using the classical fifth-order finite difference WENO discretization. We use constrained transport in order to obtain divergence-free magnetic fields, which means that we simultaneously evolve the magnetohydrodynamic (that has an evolution equation for the magnetic field) and magnetic potential equations alongside each other, and set the magnetic field to be the (discrete) curl of the magnetic potential after each time step. In this work, we compute these derivatives to fourth-order accuracy. In order to retain a single-stage, single-step method, we develop a novel Lax–Wendroff discretization for the evolution of the magnetic potential, where we start with technology used for Hamilton–Jacobi equations in order to construct a non-oscillatory magnetic field. The end result is an algorithm that is similar to our previous work Christlieb et al. (2014) [8] , but this time the time stepping is replaced through a Taylor method with the addition of a positivity-preserving limiter. Finally, positivity preservation is realized by introducing a parameterized flux limiter that considers a linear combination of high and low-order numerical fluxes. The choice of the free parameter is then given in such a way that the fluxes are limited towards the low-order solver until positivity is attained. Given the lack of additional degrees of freedom in the system, this positivity limiter lacks energy conservation where the limiter turns on. However, this ingredient can be dropped for problems where the pressure does not become negative. We present two and three dimensional numerical results for several standard test problems including a smooth Alfvén wave (to verify formal order of accuracy), shock tube problems (to test the shock-capturing ability of the scheme), Orszag–Tang, and cloud shock interactions. These results assert the robustness and verify the high-order of accuracy of the proposed scheme.

Modelling coastal water waves using a depth-integrated, non-hydrostatic model with shock-capturing ability

ABSTRACTThis paper presents a depth-integrated, non-hydrostatic model for coastal water waves. The shock-capturing ability of this model is its most attractive aspect and is essential for computation of energetic breaking waves and wet–dry fronts. The model is solved in a fraction step manner, where the total pressure is decomposed into hydrostatic and non-hydrostatic parts. The hydrostatic pressure component is integrated explicitly in the framework of the finite volume method, whereas most of the existing models use the finite difference method. The fluxes across the cell faces are computed in a Godunov-based manner through an efficient multi-stage scheme. The flow variables are reconstructed at each cell face to obtain second-order spatial accuracy. Wave breaking is treated as a shock by locally switching off the non-hydrostatic pressure in the wave front. A moving shoreline boundary is also incorporated. The robustness and accuracy of the developed model are demonstrated through numerical tests.

1D modelling of dam-break surges with floating debris

To predict flood and debris flow dynamics a numerical model, based on 1D De Saint Venant (SV) equations, was developed. The McCormack–Jameson shock-capturing scheme was employed for the solution of the equations, written in a conservative-law form. This technique was applied to determine both the propagation and the profile of a two-phase debris flow resulting from the instantaneous and complete collapse of a storage dam. To validate the model, comparisons were made between its predictions and laboratory measurements of flows of water and homogeneous granular mixtures in a uniform geometry flume that can reproduce dam-break waves. Agreements between computational and experimental results were considered very satisfactory for mature (non-stratified) debris flows, which cover most real cases. To better predict immature (stratified) flows, the model should be improved in order to feature, in a more realistic way, the distribution of the particle sizes within the mixture. On the whole, the model proposed can easily be extended to channels with arbitrary cross sections for debris flow routing, as well as for solving different problems of unsteady flow in open channels by incorporating the appropriate initial and boundary conditions. The great advantages of the technique developed are that it is based on the strong shock-capturing ability of the McCormack–Jameson numerical scheme and has a simple application of the resulting algorithm when considering 1D debris flow phenomena.