Total Variation Diminishing Scheme Research Articles

A new high-fidelity Total Variation Diminishing scheme based on the Boundary Variation Diminishing principle for compressible flows

High-accuracy and stable simulations of compressible flows have been highly demanded in many engineering situations. However, it remains a challenging task due to the complicated fluid structures in compressible flows, which include both continuous and discontinuous solutions. The total variation diminishing (TVD) scheme is a mainstream numerical method because it has second-order accuracy for the smooth solution and suppresses numerical oscillations near the discontinuous solution. The properties of the TVD scheme depend on the design of the second-order slope-limiter function, Φ(r), for which minmod, superbee, and van Leer types are commonly adopted. However, the limiter function that provides both superior numerical accuracy and stability has not yet been established. In this paper, we provide an adaptive model for determining the suitable limiter function in the second-order TVD range, unlike the conventional methods, which use one type of limiter function for the entire calculation region. To designate the suitable limiter value, we use the boundary variation diminishing (BVD) principle, which suggests the most appropriate reconstruction method within the specific candidates. The numerical results of benchmark tests show that the proposed scheme has fewer numerical dissipation errors and provides a suitable limiter value that ensures both accuracy and stability.

Propagation characteristics of tsunami waves in shear flow based on the Boussinesq model with constant vorticity

Propagation of tsunami wave in linear shear flow with constant vorticity is studied in this paper. The nonlinear and dispersive Boussinesq equations for water waves in shear flow are derived by introducing uniform vorticity into the Euler equations. Generally, the transoceanic propagation of tsunami can be well simulated by the weakly nonlinear Boussinesq model. Thus, the nonlinear and dispersive terms are retained to the first-order and second-order (ϵ,μ2) respectively in the Boussinesq equations with uniform vorticity in the present work, which satisfies the simulation accuracy especially the dispersion effects of the long-distance propagation of tsunamis and does not require huge computational cost for the simulation. The set of equations is solved with the total variation diminishing (TVD) Lax–Wendroff scheme which is second-order both in space and time. This TVD scheme is applicable to handle the dispersion effects of the N-waves and undular bores. Different patterns of tsunami wave in the water with presence of vorticity are simulated by the numerical model. Wave height and wavelength are affected by the current with vorticity. The pattern of solitary wave is significantly changed in upstream and downstream conditions. The wave deformation and propagation speed of N-waves and undular bores in the shear flow are also studied. The vorticity has impacts on the leading crest amplitude and the maximum wave appearance time for N-wave and changes the wave amplitude and the following water level for undular bores.

A unified framework for non-linear reconstruction schemes in a compact stencil. Part 1: Beyond second order

Discretization of the convection term in a three-cell compact stencil has been widely used in Computational Fluid Dynamics (CFD) of engineering because the second-order interpolation scheme is considered to be more robust and algorithmically simple than very high-order schemes. However, according to Godnunov's theorem, it has never been trivial to design an accurate and oscillation-free discretization scheme in a compact stencil. Over decades, various kinds of schemes have been proposed to increase accuracy beyond second-order without producing non-physical numerical oscillations. Representative schemes are polynomial-based non-linear interpolations such as TVD (Total Variation Diminishing), NVD (Normalized Variable Diagram) and third-order WENO (Weighted Essentially Non-oscillatory); and non-polynomial-based interpolations including RBF (Radial Basis Functions), LDLR (Local Double Logarithmic Reconstruction) and Tangent of Hyperbola for INterface Capturing (THINC). Within the semi-discretization finite volume method, existing reconstruction schemes suffer from different issues to some extent. For example, existing TVD schemes tend to cause smearing, clipping and squaring effects. Most WENO schemes and inconsistent RBF schemes are not scale-invariant. Especially, the improved WENO-Z scheme is neither scale-invariant nor mesh-size-independent. Computationally expensive non-polynomial-based interpolations usually have singularity and are dependent on tunable parameters. Thus, this work reviews reconstruction schemes by reformulating or calculating the reconstructed boundary value in the normalised-variable space where the characteristics of reconstruction schemes can be read directly. Through the accuracy analysis and the induction of existing schemes, this work proposes a Unified Normalised-variable Diagram (UND) where the following properties of a reconstruction scheme can be given diagrammatically: 1) the second-order and the third-order accuracy condition, 2) the TVD and CBC (Convection Boundedness Criterion) constrain, 3) the ENO condition, 4) the High Resolution (HR) property, 5) and the numerical dissipation and anti-dissipation error.The proposed UND framework is firstly demonstrated by the improvement of the existing schemes with a shape-preserving TVD limiter and a scale-invariant WENO scheme. Then, by defining the normalised reconstruction operator, a brand-new scheme named ROUND (Reconstruction Operator on Unified Normalised-variable Diagram) is proposed. The ROUND scheme is devised by directly configuring the normalised reconstruction operator of favourable properties on UND. This work presents a dissipative ROUND scheme in which normalised reconstruction operators are fully in the dissipative region of UND, and a low-dissipative ROUND scheme where rigorously adjusted anti-dissipation errors are introduced. Different formulations of ROUND schemes are presented to show the flexibility of designing schemes within the UND framework. The proposed dissipative and low-dissipative ROUND schemes are evaluated via benchmark tests which show the superior performance of ROUND on accuracy and resolution. Especially, in some cases, the low-dissipative ROUND schemes obtain comparable and even better resolution than classic fifth-order WENO. The significance of this work is further demonstrated by the preliminary results of applying ROUND on the FDM-IBM (Finite Difference Method and Immersed Boundary Method) framework.

Impact of a discretisation method and chemical kinetics on the accuracy of simulation of a lifted hydrogen flame

This paper presents an analysis of numerical and modelling issues based on a simulation of nitrogen-diluted hydrogen lifted flame evolving in a hot co-flow. We apply the large-eddy simulations (LES) method with the so-called ‘no combustion model’ and concentrate on the impact of chemical mechanisms and various discretisation schemes on the obtained results. The main attention is put to the latter issue in which we analyse to what extent a discretisation method of the convective terms of the scalar equations for the species and enthalpy affects the solutions. We consider eight commonly known total variation diminishing (TVD) schemes and three upwind schemes of the second order. The remaining terms in the scalar equations and all the terms in the Navier–Stokes equations are discretised applying the sixth-order compact difference method. Such an approach makes the discretisation errors of the convective terms the main factor affecting the solutions from the numerical point of view. Prior to the main analysis, the differences caused by the use of various TVD or upwind schemes are highlighted based on a single scalar transport equation with a known analytical solution. The results obtained for the flame are compared to experimental data taken from the literature. It is shown that the differences due to the application of a particular discretisation method are of similar magnitude as the differences between the simulation results and experimental data. Moreover, analysis of the impact of the chemical mechanism showed that observed differences are comparable to these originating from the use of different discretisation methods.

A realizable second-order advection method with variable flux limiters for moment transport equations

A second-order total variation diminishing (TVD) method with variable flux limiters is proposed to overcome the non-realizability issue, which has been one of major obstacles in applying the conventional second-order TVD schemes to the moment transport equations. In the present method, a realizable moment set at a cell face is reconstructed by allowing the flexible selection of the flux limiter values within the second-order TVD region. Necessary conditions for the variable flux limiter scheme to simultaneously satisfy the realizability and the second-order TVD property for the third-order moment set are proposed. The strategy for satisfying the second-order TVD property is conditionally extended to the fourth- and fifth-order moments. The proposed method is verified and compared with other high-order realizable schemes in one- and two-dimensional configurations, and is found to preserve the realizability of moments while satisfying the high-order TVD property for the third-order moment set and conditionally for the fourth- and fifth-order moments.

TVD-MOOD schemes based on implicit-explicit time integration

The context of this work is the development of first order total variation diminishing (TVD) implicit-explicit (IMEX) Runge-Kutta (RK) schemes as a basis of a Multidimensional Optimal Order detection (MOOD) approach to approximate the solution of hyperbolic multi-scale equations. A key feature of our newly proposed TVD schemes is that the resulting CFL condition does not depend on the fast waves of the considered model, as long as they are integrated implicitly. However, a result from Gottlieb et al. [1] gives a first order barrier for unconditionally stable implicit TVD-RK schemes and TVD-IMEX-RK schemes with scale-independent CFL conditions. Therefore, the goal of this work is to consistently improve the resolution of a first-order IMEX-RK scheme, while retaining its L∞ stability and TVD properties. In this work we present a novel approach based on a convex combination between a first-order TVD IMEX Euler scheme and a potentially oscillatory high-order IMEX-RK scheme. We derive and analyse the TVD property for a scalar multi-scale equation and numerically assess the performance of our TVD schemes compared to standard L-stable and SSP IMEX RK schemes from the literature. Finally, the resulting TVD-MOOD schemes are applied to the isentropic Euler equations.

A Realizable Second-Order Advection Method with Variable Flux Limiters for Moment Transport Equations

A second-order total variation diminishing (TVD) method with variable flux limiters is proposed to overcome the non-realizability issue, which has been one of major obstacles in applying the conventional second-order TVD schemes to the moment transport equations. In the present method, a realizable moment set at a cell face is reconstructed by allowing the flexible selection of the flux limiter values within the second-order TVD region. Necessary conditions for the variable flux limiter scheme to simultaneously satisfy the realizability and the second-order TVD property for the third-order moment set are proposed. The strategy for satisfying the second-order TVD property is conditionally extended to the fourth- and fifth-order moments. The proposed method is verified and compared with other high-order realizable schemes in one- and two-dimensional configurations, and is found to preserve the realizability of moments while satisfying the high-order TVD property for the third-order moment set and conditionally for the fourth- and fifth-order moments.

Comparison of finite-volume method and method of characteristics for simulating transient flow in natural-gas pipeline

In this study, two numerical methods of modeling transient flow in a single natural-gas pipeline are analyzed and compared. Specifically, the pressure-based finite-volume method (FVM)—which uses a collocated mesh with the total variation diminishing (TVD) scheme—and the conventional implicit method of characteristics (iMOC)—which uses an inertial multiplier—are considered. Network analysis is an essential part of the design and operation of natural-gas pipelines. Studies have been conducted on various numerical methods to analyze the transient flow of natural-gas pipelines, but comparative studies on the characteristics and relative advantages and disadvantages of the methods are inadequate; therefore, it is necessary to analyze and compare the underlying numerical methods to ensure their reliability. Accordingly, two test cases are considered: slow transient flow validated with experimental data, and fast transient flow with an artificial problem including valve stroking. The two methods are examined quantitatively and qualitatively. The order of accuracy, determined using the grid-convergence index, and the computational cost are evaluated for quantitative analysis. Overall, the pressure-based FVM with the TVD scheme outperforms iMOC for both slow and fast transient problems.

Impact of a discretization scheme on an autoignition time in LES of a reacting droplet-laden mixing layer

We analyse an autoignition process in a two-phase flow in a temporally evolving mixing layer formed between streams of a cold liquid fuel (heptane at 300 K) and a hot oxidizer (air at 1000 K) flowing in opposite directions. We focus on the influence of a discretization method on the prediction of the autoignition time and evolution of the flame in its early development phase. We use a high-order code based on the 6th order compact difference method for the Navier–Stokes and continuity equations combined with the 2nd order Total Variation Diminishing (TVD) and 5th order Weighted Essentially Non-Oscillatory (WENO) schemes applied for the discretization of the advection terms in the scalar transport equations. The obtained results show that the autoignition time is more dependent on the discretization method than on the flow initial conditions, i.e., the Reynolds number and the initial turbulence intensity. In terms of mean values, the autoignition occurs approximately 15% earlier when the TVD scheme is used. In this case, the ignition phase characterizes a sharp peak in the temporal evolution of the maximum temperature. The observed differences are attributed to a more dissipative character of the TVD scheme. Its usage leads to a higher mean level of the fuel in the gaseous form and a smoother distribution of species resulting in a lower level of the scalar dissipation rate, which facilitates the autoignition process.

A novel finite-volume TVD scheme to overcome non-realizability problem in quadrature-based moment methods

A new finite-volume total variation diminishing (TVD) scheme is proposed for the solution of moment transport equations in quadrature-based moment methods (QBMM). The proposed scheme is capable of preserving important properties of the moments, such as realizability and boundedness. The idea behind the approach is to limit the flux of all the moments at each cell face with the same limiter value. The proposed numerical technique is eventually compared with other realizable schemes developed for the moment transport equations, showing that the method is able to keep the moments realizable and bounded at the same time.

High order TVD scheme for hyperbolic conservation laws

In this paper, we introduced a new third order finite difference scheme for solving initial value problems for hyperbolic conservation laws. This method is a finite difference scheme combining a dissipative scheme (diffusion scheme) and a dispersive scheme (leap frog scheme) to get a dispersive scheme with less spurious oscillations. The new scheme has the following advantages: it is a third order accuracy in space and time, simple to implement, it has the lowest order of dissipation which reduces the spurious oscillations generated by the numerical methods. The new scheme is proved stable for initial boundary value problems for linear and nonlinear scalar problems. The total variation diminishing (TVD) technique of making the third order scheme oscillations free is carried out. The extension of the scheme to to nonlinear systems of equations is presented. To validate the performance of the proposed scheme, a numerical experiments with one-and two-dimensional problems are presented and compared with exact solutions and other methods.

TVDal: Total variation diminishing scheme with alternating limiters to balance numerical compression and diffusion

Total variation diminishing (TVD) advection schemes are widely used in ocean modelling. Due to the constraints of flux limiters, TVD schemes with common single flux limiters of second-order or third-order accuracy exhibit defects associated with numerical compression or diffusion. To reduce the numerical errors induced by these numerical defects, a TVD method employing alternating flux limiters (TVDal) is proposed. The principle of TVDal is to alternately use a compressive and a diffusive limiter in different time steps. Through balancing the compression of the Superbee limiter and the diffusion of other flux limiters, a reduction of ∼60% of numerical errors in one-dimensional benchmark tests can be achieved with the TVDal scheme. In addition, the TVDal scheme combined with the Strang-splitting method is proposed for multi-dimensional advection. For an idealized 2D experiment with oblique advection, the Strang-splitting TVDal scheme features good shape retention with much lower numerical errors. The TVDal scheme has also been applied to model tidally induced internal lee waves, indicating that TVDal has the potential ability to balance the numerical compression and diffusion of conventional TVD schemes for cases with highly nonlinear, stratified, and nonhomogeneous flow fields. Finally, the principles of the TVDal scheme also show the potential application to conventional high-order TVD schemes.

A Numerical Study on Jet Structure and Noise Emission of Underexpanded Radial Jet

In this study, an underexpanded radial jet issuing from a small gap between two circular tubes facing each other is investigated numerically. Radial jet is formed, for example, downstream of high-pressure valves in piping system and of poppet valves in engines, and causes many industrial problems such as the noise generation and the fatigue failure of structure. In this study, the jet issuing from a small gap between two tubes with same diameter is numerically simulated. The flow field is assumed to be axisymmetric against the central axis of tubes and to be symmetric against the intermediate plane between the exits of two tubes. The axisymmetric Euler equations are solved using symmetric TVD (Total Variation Diminishing) scheme. The effects of nozzle pressure ratio and of diameter of circular tubes on the structure and the behavior of jets are examined. Typical cell structure of underexpanded jet appears in radial jet and the length of cell becomes smaller in downstream region because the jet spreads radially like a disc. The length and width of first cell are larger with higher nozzle pressure ratio. Many vortices are generated one after another near the jet boundary and move downstream, which cause the oscillation of jet. Outside of jet, two types of density waves are observed. One of them propagates toward the nozzle (toward the upstream region) and the other propagates in opposite direction. Focusing on the pressure change caused by the former waves, which is related to well-known screech, dominant frequency obtained by FFT analysis was found to become lower with higher pressure ratio and smaller diameter of tube.

Numerical Modeling and Validation of a Novel 2D Compositional Flooding Simulator Using a Second-Order TVD Scheme

The aim of this paper is to present the latter and develop a numerical simulator aimed at solving a 2D domain porous medium, using the compositional approach to simulate chemical flooding processes. The simulator consists in a two-phase, multicomponent system solved by the IMplicit in Pressure, Explicit in Concentration (IMPEC) approach, which can be operated under an iterative/non-iterative condition on each time-step. The discretization of the differential equations is done using a fully second order of accuracy, along with a Total Variation Diminishing (TVD) scheme with a flux limiter function. This allowed reducing the artificial diffusion and dispersion on the transport equation, improving the chemical species front tracking, decreasing the numerical influence on the recovery results. The new model was validated against both commercial and academic simulators and moreover, the robustness and stability were also tested, showing that the iterative IMPEC is fully stable, behaving as an implicit numerical scheme. The non-iterative IMPEC is conditionally stable, with a critical time-step above which numerical spurious oscillations begin to appear until the system numerically crashes. The results showed a good correspondence in different grid sizes, being largely affected by the time-step, with caused a decrease in the recovery efficiency in the iterative scheme, and the occurrence of numerical oscillations in the non-iterative one. Numerically speaking, the second-order scheme using a flux splitting TVD discretization proved to be a good approach for compositional reservoir simulation, decreasing the influence of numerical truncation errors on the results when compared to traditional, first-order linear schemes. Along with these studies, secondary recoveries in constant and random permeability fields are simulated before employing them in tertiary recovery processes.

Simulating Laboratory Braided Rivers with Bed-Load Sediment Transport

Numerical models provide considerable assistance in the investigation of complicated processes in natural rivers. In the present study, a physics-based two-dimensional model has been developed to simulate the braiding processes and morphodynamic changes in braided rivers. The model applies the basic hydrodynamic and sediment transport principles with bed morphology deformation and a TVD (Total Variation Diminishing) scheme to predict trans-critical flows and bed morphology deformation. The non-equilibrium transport process of graded bed load sediment is simulated, with non-uniform sediments, secondary flows, and sheltering effects being included. A multiple bed layer technique is adopted to represent the vertical sediment sorting process. The model has been applied to simulate the bed evolution process in an experimental river with bed load transport. Comparisons between the experimental river and predicted river are analysed, including their pattern evolution processes, important braiding phenomena, and statistical characteristics. Avulsion activities have been found in the braiding evolution process, representing the primary ways in which channels form and disappear in braided rivers. The increases in the active braiding intensity and total braiding intensity show similar trends to those observed in the experimental river. Statistical methods are applied to assess the scale-invariant topographic properties of the simulated river and real rivers. The model demonstrated its potential to predict the morphodynamics in natural rivers.

Implementation and Comparison of High-Resolution Spatial Discretization Schemes for Solving Two-Fluid Seven-Equation Two-Pressure Model

As compared to the two-fluid single-pressure model, the two-fluid seven-equation two-pressure model has been proved to be unconditionally well-posed in all situations, thus existing with a wide range of industrial applications. The classical 1st-order upwind scheme is widely used in existing nuclear system analysis codes such as RELAP5, CATHARE, and TRACE. However, the 1st-order upwind scheme possesses issues of serious numerical diffusion and high truncation error, thus giving rise to the challenge of accurately modeling many nuclear thermal-hydraulics problems such as long term transients. In this paper, a semi-implicit algorithm based on the finite volume method with staggered grids is developed to solve such advanced well-posed two-pressure model. To overcome the challenge from 1st-order upwind scheme, eight high-resolution total variation diminishing (TVD) schemes are implemented in such algorithm to improve spatial accuracy. Then the semi-implicit algorithm with high-resolution TVD schemes is validated on the water faucet test. The numerical results show that the high-resolution semi-implicit algorithm is robust in solving the two-pressure two-fluid two-phase flow model; Superbee scheme and Koren scheme give two highest levels of accuracy while Minmod scheme is the worst one among the eight TVD schemes.

High order accurate dual-phase-lag numerical model for microscopic heating in multiple domains

In this article, a characteristic-based dual-phase-lag numerical model based on finite difference method has been developed to predict the microscopic heating response in time as well as consideration of the micro-structured effect. High-order TVD (Total Variation Diminishing) schemes being oscillation-free can yield high-order accurate solutions without introducing wiggles and therefore are utilised in this work. A multi-domain approach integrated within the dual-phase-lag numerical model allows the computation of microscopic conjugate heat transfer problems. Effects of different phase-lag values on the behaviour of heat transfer are investigated. The model is capable of predicting temperature patterns transiting from the wave nature of heat propagation to additional diffusion being experienced within different solid regions via phonon–electron interaction or phonon scattering.

A solver for the two-phase two-fluid model based on high-resolution total variation diminishing scheme

Finite volume techniques with staggered mesh are used to develop a new numerical solver for the one-dimensional two-phase two-fluid model using a high-resolution, Total Variation Diminishing (TVD) scheme. The solver is implemented to analyze numerical benchmark problems for verification and testing its abilities to handle discontinuities and fast transients with phase change. Convergence rates are investigated by comparing numerical results to analytical solutions available in literature for the case of the faucet flow problem. The solver based on a new TVD scheme is shown to exhibit higher-order of accuracy compared to other numerical schemes. Mass errors are also examined when phase change occurs for the shock tube problem, and compared to those of the 1st-order upwind scheme implemented in the nuclear thermal-hydraulics code TRACE. The solver is shown to exhibit numerical stability when applied to problems with discontinuous solutions and results of the new solver are free of spurious oscillations.

Large Time Step TVD Schemes for Hyperbolic Conservation Laws

Large time step (LTS) explicit schemes in the form originally proposed by LeVeque [Comm. Pure Appl. Math., 37 (1984), pp. 463--477] have seen a significant revival in recent years. In this paper we consider a general framework of local (2k+1)-point schemes containing LeVeque's scheme (denoted as LTS-Godunov) as a member. A modified equation analysis allows us to interpret each numerical cell interface coefficient of the framework as a partial numerical viscosity coefficient. We identify the least and most diffusive total variation diminishing (TVD) schemes in this framework. The most diffusive scheme is the(2k+1)-point Lax--Friedrichs scheme (LTS-LxF). The least diffusive scheme is the LTS scheme of LeVeque based on Roe upwinding (LTS-Roe). Herein, we prove a generalization of Harten's lemma: all partial numerical viscosity coefficients of any local unconditionally TVD scheme are bounded by the values of the corresponding coefficients of the LTS-Roe and LTS-LxF schemes. We discuss the nature of entropy violations associated with the LTS-Roe scheme; in particular we extend the notion of transonic rarefactions to the LTS framework. We provide explicit inequalities relating the numerical viscosities of LTS-Roe and LTS-Godunov across such generalized transonic rarefactions and discuss numerical entropy fixes. Finally, we propose a one-parameter family of LTS TVD schemes spanning the entire range of the admissible total numerical viscosity. Extensions to nonlinear systems are obtained through the Roe linearization. The one-dimensional Burgers equation and the Euler system are used as numerical illustrations.

A conservative finite volume scheme with time-accurate local time stepping for scalar transport on unstructured grids

In this article we propose a new conservative high resolution TVD (total variation diminishing) finite volume scheme with time-accurate local time stepping (LTS) on unstructured grids for the solution of scalar transport problems, which are typical in the context of water quality simulations. To keep the presentation of the new method as simple as possible, the algorithm is only derived in two space dimensions and for purely convective transport problems, hence neglecting diffusion and reaction terms. The new numerical method for the solution of the scalar transport is directly coupled to the hydrodynamic model of Casulli and Walters (2000) that provides the dynamics of the free surface and the velocity vector field based on a semi-implicit discretization of the shallow water equations. Wetting and drying is handled rigorously by the nonlinear algorithm proposed by Casulli (2009). The new time-accurate LTS algorithm allows a different time step size for each element of the unstructured grid, based on an element-local Courant-Friedrichs-Lewy (CFL) stability condition. The proposed method does not need any synchronization between different time steps of different elements and is by construction locally and globally conservative. The LTS scheme is based on a piecewise linear polynomial reconstruction in space–time using the MUSCL–Hancock method, to obtain second order of accuracy in both space and time.The new algorithm is first validated on some classical test cases for pure advection problems, for which exact solutions are known. In all cases we obtain a very good level of accuracy, showing also numerical convergence results; we furthermore confirm mass conservation up to machine precision and observe an improved computational efficiency compared to a standard second order TVD scheme for scalar transport with global time stepping (GTS). Then, the new LTS method is applied to some more complex problems, where the new scalar transport scheme has also been coupled to a semi-implicit model for the simulation of the free surface hydrodynamics, including nonlinear wetting and drying. The last application shown in this paper is carried out on a real geometry, for which we have taken available DTM data of the lake Guaiba in Brazil. Comparisons have been made in all cases with a second order TVD scheme based on GTS. For the new LTS algorithm we report a significant reduction in computational effort, with a savings of CPU time of the order of up to 95%.