Minmod Limiter Research Articles

Resolution of the bidimensional shallow water equations with horizontal temperature gradients by a well Balanced Roe scheme

This work presents a numerical resolution of the two-dimensional Ripa system using Roe scheme. A discretization of the source term satisfying the conservation property is presented. The mathematical model is based on the ordinary shallow water equations in merging the horizontal temperature gradient. The numerical approach employs unstructured grids and integrates Runge–Kutta method and the minmod limiter to achieve second order accuracy in both time and space domains. A strategy of mesh adaptation based on temperature gradient is implemented to refine the study domain and gain in computational cost. Various numerical tests are conducted to verify the efficiency of the numerical method.

Low-dissipation central-upwind schemes for compressible multifluids

We introduce second-order low-dissipation (LD) path-conservative central-upwind (PCCU) schemes for the one- (1-D) and two-dimensional (2-D) multifluid systems, whose components are assumed to be immiscible and separated by material interfaces. The proposed LD PCCU schemes are derived within the flux globalization based PCCU framework and they employ the LD central-upwind (LDCU) numerical fluxes. These fluxes have been proposed in [A. Kurganov and R. Xin, J. Sci. Comput., 96 (2023), Paper No. 56] for the single-fluid compressible Euler equations, recently modified in [S. Chu, A. Kurganov, and R. Xin, submitted], and in this paper, we rigorously develop their multifluid extensions. In order to achieve higher resolution near the material interfaces, we track their locations and use an overcompressive SBM limiter in their neighborhoods, while utilizing a dissipative generalized minmod limiter in the rest of the computational domain. We first develop a second-order finite-volume LD PCCU scheme and then extend it to the fifth order of accuracy via the finite-difference alternative weighted essentially non-oscillatory (A-WENO) framework. We apply the developed schemes to a number of 1-D and 2-D numerical examples to demonstrate the performance of the new schemes.

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Simulation of Sloped-Bed Tuned Liquid Dampers Using a Nonlinear Shallow Water Model

Energy dissipative and positivity preserving schemes for large-convection ion transport with steric and solvation effects

Ion transport plays a crucial role in biophysical and electrochemical applications. While being modeled by the classical Poisson–Nernst–Planck (PNP) equations, many ionic features are ignored in the course-graining modeling of ionic interactions. In this work, we study ion transport by modified PNP equations including effects arising from steric interactions and the Born solvation, which both could cause strong convection. The modified NP equations are first rewritten in a gradient-flow reformulation that involves mobility and chemical potential terms. The mobility is then approximated by upwind fluxes with various flux limiters on the half-grid points, and the chemical potential terms are then implicitly approximated by central differencing. The resulting first-order in time scheme is proved to be energy dissipative unconditionally. Furthermore, the unconditional existence of positive numerical concentration is proved by making use of the singularity of the entropy term, which prevents the concentration from approaching zero. The flux with the Minmod or Osher limiter is reduced to a second-order upwind flux in the strong convective limit. Numerical tests are performed to confirm the expected order of accuracy and the properties of the developed scheme at discrete level. In addition, our numerical experiments demonstrate that the developed schemes with limiters can achieve higher accuracy than the Scharfetter-Gummel scheme and classical upwind scheme in tests with strong convection. Finally, the developed schemes are applied to numerically assess the effects of Coulombic interaction, steric interaction, and Born solvation on the rectifying transport of ions through nanopores.

Stability effect of multidimensional velocity components in numerical flux SLAU

AbstractIn computational fluid dynamics of compressible fluid flow, the simple low‐dissipation advection upstream (SLAU) scheme formulated with multidimensional velocity components (normal and parallel to a cell interface) is a widely employed all‐speed scheme. As a variant of SLAU, the mSLAU scheme, which adopts only a velocity component normal to the cell interface instead of multidimensional velocity components, is used for rotorcraft calculations. However, although mSLAU has been claimed to be empirically stable, it has been pointed out that using only the cell‐interface‐normal velocity component instead of the multidimensional velocity components causes numerical instability. Therefore, to clarify the roles of the multidimensional velocity components for computational stability, we solved some benchmark problems associated with using SLAU or mSLAU. We discovered that the multidimensional velocity components contributed to stability against poor‐quality grids by isotropically producing enough amount of numerical dissipation, especially in low‐subsonic and supersonic flows. Although mSLAU could practically treat moderate Mach number flows (approximately 0.1 < M < 1.0) when coupled with the minmod limiter, using only the cell‐interface‐normal velocity component can deteriorate convergence of calculations and lead to susceptibility in the grid geometry.

A spectral MUSCL scheme for gPC-Galerkin method to uncertain hyperbolic equations

This paper deals with a problem that we found when directly implementing the scalar MUSCL scheme to spectral coefficient equations of the gPC-Galerkin method. The order degradation at extrema of spectral coefficients and non-uniform reconstruction process lead to a distortion of standard deviation at some extrema, especially where the peak consists of several extrema of coefficients. From the perspective of probability space, a spectral MUSCL scheme is proposed based on the generalized minmod limiter to avoid the defects mentioned above when applying the scalar MUSCL. In this paper, we present some properties of the new scheme related to total variation and demonstrate them by the uncertain linear scalar conservation law. Finally, we employ this scheme to compressible Euler equations, and a good correlation of standard deviation is obtained. The details of implementation of the gPC-Galerkin method for Euler equations are also discussed, and the approximate Jacobian matrix is adopted.

Effects of Different Slope Limiters on Stratified Shear Flow Simulation in a Non-hydrostatic Model

To simulate the dynamical structures of stratified shear flows, the high-resolution Total Variation Diminishing (TVD) method is necessary and widely-used due to its high-order spatial accuracy, oscillation control, and ability to capture the well-defined structures of vortices. Lack of understanding the TVD slope limiters usually results in inaccurate numerical simulation on stratified shear flows in terms of shear instability and spatiotemporal variations of mixing. In this study, the performances of four typical TVD slope limiters, namely the minmod, van Leer, Monotonized Central (MC), and superbee limiters, were investigated on modelling stratified shear flows based on the open-source non-hydrostatic model, NHWAVE. The four slope limiters are all commonly-used and have the typical numerical characteristics. All the limiters were respectively applied in two classical test cases, namely, shear instability and lock-exchange problem. The simulation results showed that the effects of slope limiters were correlated with their characteristics of numerical dissipation (or anti-dissipation), which can influence notably the model predictions of the generation of shear instability, the development of interfacial structures, and the mixing process. In the test cases, MC limiter’s performance was the best, because it could simulate the well-defined structures of instability while not introducing noticeable error. Minmod has an excessively large dissipation, which introduced noticeable numerical errors that can influence the model accuracy and can even suppress or omit the generation of interfacial vortices. Superbee limiter, the most anti-dissipative one, usually over-predicted the instability and mixing effects in time and space domain, and was likely to cause computational instability in some cases. The performances of van Leer and MC were similar, but their predictions of the evolutions of interfacial structures and mixing could be significantly different. Besides, the co-effects of grid resolution and slope limiters were also investigated; it was found that the refinement of grids may not help to reproduce a higher-quality result with a specific slope limiter.

Development and Application of a High-Speed Reacting Flow Solver in OpenFOAM

A numerical study was conducted to verify and validate a new solver, named hyperReactingSodFoam, based on the OpenFOAM toolbox. This study mainly focuses on verification of solver’s computational capability to compute hypersonic flows involving shock-induced combustion, and its applications. This investigation is limited to inviscid flow and incorporates Minmod limiter to achieve total variation diminishing property. Analysis of a practical case was conducted by simulate and validate hypersonic blunt body projectile experiments at Mach 6.46 and 4.18. Likewise, to demonstrate the wider applicability of solver, a standing detonation in an oblique detonation wave engine combustor with a sharp as well as a blunted wedge geometry was studied. It was observed that, for a given flow condition and combustor geometry, a sharp wedge has a better mass-weighted average pressure recovery comparative to the blunted wedge. However, blunted wedge geometry provides a better picture of the detonation phenomenon as a practical wedge will have finite bluntness. In addition, the concept of the detonation induction length becomes less relevant for a blunted wedge.

Numerical modeling of two dimensional non-capacity model for sediment transport by an unstructured finite volume method with a new discretization of the source term

The main goal of this work is the resolution of the two-dimensional shallow water equations of water–sediment mixture coupled to the transport diffusion equation for the total sediment load, and bed change rate equation by Roe scheme with a new discretization of the source term. The proposed discretization is well-balanced with the flux gradient and uses data right and left on the interfaces between two control volumes and satisfies the C-property. The numerical method uses unstructured meshes and incorporates minmod limiter and Runge–Kutta method to reach second order spatial and temporal accuracy. We also use an adaptive mesh based on gradient concentration of sediments to refine the study domain with a lower computational cost. We present some numerical results in order to verify and validate the performance of the numerical scheme, particular attention is given to the treatment of the dam-break problem over mobile beds. The numerical scheme demonstrates the intended accuracy and robustness to modelize dam-break flows over erodible bed.

Wave propagation in generalized thermo-poro-elastic media via wavelet-based cell-adaptive central high resolution schemes using UNO limiters

The uniformly non-oscillatory (UNO) limiters with second-order accuracy are updated to handle nonuniform cell obtained from wavelet-based cell adaptation. These limiters are then used in the cell-adaptive second-order Kurganov-Tadmor (KT) central high resolution scheme. Over the adapted cells, the performance of the UNO limiters are compared with the generalized MINMOD (GMM) limiter with the total variation diminishing (TVD) feature, by measuring error of reconstruction of some functions and also by numerical solution of some benchmark partial differential equations (PDEs). Then, unified forms of generalized fully-coupled saturated thermo-poro-elastic systems are presented as first-order hyperbolic-parabolic systems in the 2D Cartesian and the asymmetric spherical coordinates. For these systems, corresponding flux, source (load), diffusion and nonlinear terms are derived. It is shown that in the case of the asymmetric spherical coordinate, due to the geometry, considerable complexities are introduced in the load (source) and flux terms. Finally, the coupled 1D and 2D systems are simulated by the adaptive central KT scheme with the GMM and UNO limiters.

A strong coupling prediction-correction immersed boundary method

In the traditional immersed boundary methods for solving compressible fluid-structure interaction problems, conservation is one of the problems that must be considered. When the coupling boundary moves on the fixed grid, the structure coverage will change, resulting in many dead elements and emerging elements on the fluid grid. In the ghost-cell immersed boundary method, the reconstructed grid can not maintain the strict mass conservation when the dead elements and emerging elements appear. In order to overcome the shortcomings of traditional methods, a strong coupling prediction-correction immersed boundary method for compressible fluid-structure interaction problems was proposed. Firstly, the matrix representation of a general fluid-structure coupling system was described, and a strong coupling Gauss-Seidel iterative scheme of fluid-structure coupling system was derived. Furthermore, a prediction-correction scheme was derived, and a prediction-correction immersed boundary method was proposed. The fluid-structure coupling boundary was regarded as a free surface, and the space originally occupied by the solid was initialized as zero mass elements, allowing the fluid to pass through the coupling boundary freely. For the calculation of fluid, the second-order MUSCL finite volume scheme with the minmod limiter and the AUSM+-up flux based on Zha-Bilgen splitting were used to advance the time step with the third-order Runge-Kutta scheme. In the correction step, the transport process was realized by a set of mass conservation transport rules. The transport algorithm could be summarized as marking the fluid inside the boundary, dividing and moving the fluid in a uniform way according to the marking order, generating a flow pointing to the outside of the boundary, and finally applying a velocity correction near the boundary to ensure the no-slip condition. The marking and transport algorithm avoided the tedious geometric treatment of the cut-cells, and ensured the easy implementation of the algorithm. For the calculation of solids, the first-order difference scheme and the implicit dynamic finite element scheme were used to solve the rigid body and linear elastic body respectively, and the Gauss quadrature was used to obtain the coupling force on the solid surface. The one-dimensional and two-dimensional problems were calculated by the prediction-correction immersed boundary method. In the one-dimensional piston problem, the accuracy, conservation and convergence of the method were investigated by comparing the results with those in the literature. In the two-dimensional shock wave impact problem, the experimental optical schlieren images were compared with those obtained by the numerical simulation, and the deflection history of the plate structure was investigated. The study shows that this method can accurately maintain the mass conservation of the flow field and has the advantage of easy implementation, which is different from the traditional ghost-cell method and the cut-cell method. This method has the first-order convergence accuracy, and can accurately predict the flow field after shock diffraction and the deflection of plate under shock waves. It provides a new idea for the development of fluid-structure coupling algorithms.

Evaluating the influence of slope limiters on nearshore wave simulation in a non-hydrostatic model

In a non-hydrostatic model utilizing a Godunov-type finite volume method, the numerical dissipation introduced by slope limiters can be a main source of total dissipation. The objective of this study is to look into how different slope limiters influence nearshore wave simulation with an emphasis on depth-limited wave breaking simulation in the open-source NHWAVE model. The minmod, superbee and van Leer limiter are analyzed. The performances of different slope limiters are correlated with their characteristics. The refinement of the grids do not necessarily help to improve the model accuracy. In the shock-capturing modeling of solitary waves, although superbee limiter introduces a small non-physical increase of wave energy before wave breaking, the overall dissipation is largest. The van Leer limiter does not introduce noticeable dissipation before breaking in a relatively short simulation time, and the energy dissipated is moderate of all three limiters. In the turbulent modeling of wave breaking, slope limiters do not introduce large numerical dissipation and the model accuracy is mainly influenced by the dissipation in wave propagation and generating turbulent dissipation at different orders of magnitude. The superbee limiter predicts the largest turbulent kinetic energy at breaking moment, which means superbee limiter tends to reproduce the most intensive breaking. It is consistent with the largest wave energy dissipation in the shock-capturing modeling.

Чисельне моделювання нестаціонарних потоків холодної плазми при роботі плазмового актуатора

The numerical simulation of unsteady flows of cold plasma is considered in this article. A low-temperature non-equilibrium ideal plasma is formed when the plasma actuator interacts with the air. The mathematical model has been developed to describe the behavior of low-temperature plasma. It is based on non-stationary equations describing the dynamics of charged particles and plasma electrodynamics equations. The 14 types of particles: metastable and excited nitrogen and oxygen atoms, positive and negative ions, electrons and atomic oxygen are considered. Volumetric and surface chemical reactions describing processes in a barrier discharge that occur above the dielectric surface are considered. For non-stationary equations of plasma dynamics, an implicit numerical algorithm with pseudo-time iteration has been developed, which is based on a finite-volume approach. The equation for the electrostatic potential with sources was solved using the generalized minimal residual method with incomplete LU preconditioning. In non-stationary equations for the density of plasma particles, the drift derivatives were approximated using the TVD scheme with the MinMod limiter function. The derivatives in the equation for the electric potential were calculated using finite-volume relations taking into account the upwind approximation of the concentration of charged plasma particles. The numerical results of the generation, propagation and destruction of a streamer during a dielectric barrier discharge are obtained. The unsteady plasma characteristics in the region above the dielectric surface are analyzed, including the distribution of the particles density, electric potential and the Lorentz force components. The results of numerical simulation of unsteady flows of low-temperature plasma are in good agreement with the available experimental data.

Vorticity Confinement Applied to Accurate Prediction of Convection of Wing Tip Vortices and Induced Drag

The vorticity confinement (VC) method was applied to tip vortices shed by edges of wings in order to predict induced drag using far-field integration. The optimal VC parameter was determined by its application to tip vortices shed by 3-D stationary and rotating wings. The VC was used with a total variation diminishing (TVD) approach to reduce the needed confinement. The TVD minmod and differentiable flux limiters were tested to evaluate their effect on the VC parameter. The 3-D inviscid simulations were post-processed using the wake-integral technique to determine lift-induced drag force. Dependence of the VC parameter on the flight Mach number and the angle of attack was evaluated. Grid convergence studies were conducted for induced drag generated by 3-D wing. The VC approach was combined with the Reynolds stress equation turbulence model, and the results were compared to experimental data of tip vortex evolution.

Entropy stable adaptive moving mesh schemes for 2D and 3D special relativistic hydrodynamics

This paper develops entropy stable (ES) adaptive moving mesh schemes for the 2D and 3D special relativistic hydrodynamic (RHD) equations. They are built on the ES finite volume approximation of the RHD equations in curvilinear coordinates, the discrete geometric conservation laws, and the mesh adaptation implemented by iteratively solving the Euler-Lagrange equations of the mesh adaption functional in the computational domain with suitably chosen monitor functions. First, a sufficient condition is proved for the two-point entropy conservative (EC) flux, by mimicking the derivation of the continuous entropy identity in curvilinear coordinates and using the discrete geometric conservation laws given by the conservative metrics method. Based on such sufficient condition, the EC fluxes for the RHD equations in curvilinear coordinates are derived and the second-order accurate semi-discrete EC schemes are developed to satisfy the entropy identity for the given convex entropy pair. Next, the semi-discrete ES schemes satisfying the entropy inequality are proposed by adding a suitable dissipation term to the EC scheme and utilizing linear reconstruction with the minmod limiter in the scaled entropy variables in order to suppress the numerical oscillations of the above EC scheme. Then, the semi-discrete ES schemes are integrated in time by using the second-order strong stability preserving explicit Runge-Kutta schemes. Finally, several numerical results show that our 2D and 3D ES adaptive moving mesh schemes effectively capture the localized structures, such as sharp transitions or discontinuities, and are more efficient than their counterparts on uniform mesh.

Combined Vorticity Confinement and TVD Approaches for Accurate Vortex Modelling

The vorticity confinement (VC) method was used with total variation diminishing (TVD) schemes to reduce possible over-confinement and applied to convected and stationary vortices. The optimal VC parameter was determined by application to 2-D vortices. The conservativity of the scheme is investigated in terms of kinetic energy and momentum. Grid convergence study was conducted for the range of grid steps. The log–log plot of the error shows that the second-order upwind scheme with the VC nearly reaches the 4th order of accuracy for the wide range of grid cell sizes. VC was used with TVD minmod and differentiable flux limiters to evaluate their effect on the VC method and prevention of over-confinement.

A staggered semi-implicit hybrid FV/FE projection method for weakly compressible flows

In this article we present a novel staggered semi-implicit hybrid finite-volume/finite-element (FV/FE) method for the resolution of weakly compressible flows in two and three space dimensions. The pressure-based methodology introduced in [1,2] for viscous incompressible flows is extended here to solve the compressible Navier-Stokes equations. Instead of considering the classical system including the energy conservation equation, we replace it by the pressure evolution equation written in non-conservative form. To ease the discretization of complex spatial domains, face-type unstructured staggered meshes are considered. A projection method allows the decoupling of the computation of the density and linear momentum variables from the pressure. Then, an explicit finite volume scheme is used for the resolution of the transport diffusion equations on the dual mesh, whereas the pressure system is solved implicitly by using continuous finite elements defined on the primal simplex mesh. Consequently, the CFL stability condition depends only on the flow velocity, avoiding the severe time restrictions that might be imposed by the sound velocity in the weakly compressible regime. High order of accuracy in space and time of the transport diffusion stage is attained using a local ADER (LADER) methodology. Moreover, also the CVC Kolgan-type second order in space and first order in time scheme is considered. To prevent spurious oscillations in the presence of shocks, an ENO-based reconstruction, the minmod limiter or the Barth-Jespersen limiter are employed. To show the validity and robustness of our novel staggered semi-implicit hybrid FV/FE scheme, several benchmarks are analysed, showing a good agreement with available exact solutions and numerical reference data from low Mach numbers, up to Mach numbers of the order of unity.

Efficient approach toward the application of the Godunov method to hydraulic transients

Abstract The proposed study investigated the applicability of the finite volume method (FVM) based on the Godunov scheme to transient water hammer with shock front simulation, in which intermediate fluxes were computed using either first-order or second-order Riemann solvers. Finite volume (FV) schemes are known to conserve mass and momentum and produce the efficient and accurate realization of shock waves. The second-order solution of the Godunov scheme requires an efficient slope or a flux limiter for error minimization and time optimization. The study examined a range of limiters and found that the MINMOD limiter is the best for modeling water hammer in terms of computational time and accuracy. The first- and second-order FVMs were compared with the method of characteristics (MOCs) and experimental water hammer measurements available in the literature. Both the FV methods accurately predicted the numerical and experimental results. Parallelization of the second-order FVM reduced the computational time similar to that of first-order. Thus, the study presented a faster and more accurate FVM which is comparable to that of MOC in terms of computational time and precision, therefore it is a good substitute for the MOC. The proposed study also investigated the implementation of a more complex convolution-based unsteady friction model in the FVM to capture real pressure dissipation. The comparison with experimental data proved that the first-order FV scheme with the convolution integral method is highly accurate for computing unsteady friction for sudden valve closures.

Compressive properties of Min-mod-type limiters in modelling shockwave-containing flows

The long-ignored compressive properties of Min-mod-type limiter is investigated in this manuscript by demonstrating its potential in numerically modelling shockwave-containing flows, especially in shock wave/boundary layer interaction (SWBLI) problems. Theoretical studies were firstly performed based on Sweby’s total variation diminishing (TVD) limiter region and Spekreijse’s monotonicity-preserving limiter region to indicate Min-mod-type limiters’ compressive properties. The influence of limiters on the solution accuracy was evaluated using a hybrid-order analysis method based on the grid-independent study in three typical shockwave-containing flows. The conclusions are that, Min-mod-type limiter can be utilized as a dissipative and/or compressive limiter, but depending on the reasonable value of the compression parameter. The compressive Min-mod limiter tends to be more attractive in modelling shockwave-containing flows as compared to other commonly preferred limiters because of its stable computational process and its high-resolution predictions. However, the compressive Min-mod limiter may suffer from its slightly poor convergence, as that observed in other commonly accepted smooth limiters in modelling SWBLI problems.

A positivity-preserving Lagrangian Discontinuous Galerkin method for ideal magnetohydrodynamics equations in one-dimension

In this paper, we propose a conservative Lagrangian Discontinuous Galerkin (DG) scheme for solving the ideal compressible magnetohydrodynamics (MHD) equations in one-dimensional. This scheme can preserve positivity of physically positive variables such as density and thermal pressure. We first develop a Lagrangian HLLD approximate Riemann solver which can keep positivity-preserving property under some appropriate signal speeds. With this solver a first order positivity-preserving Lagrangian DG scheme can be constructed. Then we design a high order positivity-preserving and conservative Lagrangian DG scheme by using the strong stability preserving (SSP) high order time discretizations and the positivity-preserving scaling limiter. We adopt TVB minmod limiter with the local characteristic fields for the ideal MHD system to control spurious oscillations around the shock wave. Some numerical examples are presented to demonstrate the accuracy and positivity-preserving property of our scheme.