Second-order Total Variation Diminishing Research Articles

On the accuracy of calculating invariants in centered rarefaction waves and in their influence area

We perform a comparative analysis of the accuracy of second-order TVD (Total Variation Diminishing), third-order RBM (Rusanov-Burstein-Mirin), and fifth-order in space and third-order in time A-WENO (Alternative Weighted Essentially Non-Oscillatory) difference schemes for solving a special Cauchy problem for shallow water equations with discontinuous initial data. The exact solution of this problem contains a centered rarefaction wave and does not contain a shock wave. It is shown that in the centered rarefaction wave and its influence area, the solutions of these three schemes with different orders converge to different invariants of the exact solution. This leads to a decrease in the accuracy of these schemes when calculating the vector of base variables of the considered Cauchy problem. The P-form of the first differential approximation of the difference schemes is used for the theoretical justification of these numerical results.

LES of autoignition process in a turbulent droplet-laden mixing layer – impact of an evaporation model and discretisation method

This paper focuses on the numerical modelling of the autoignition process in an ethanol-air mixture within a turbulent droplet-laden temporally evolving mixing layer, utilising the Large Eddy Simulation (LES) method. We compare results obtained by implementing two notably distinct discretisation methods (second-order Total Variation Diminishing – TVD, and fifth-order Weighted Essentially Non-Oscillatory – WENO) in the species and energy transport equations. Additionally, we examine three widely recognised evaporation models (the ‘ D 2 ’ model, the Abramzon–Sirignano model, and a non-equilibrium model based on the Langmuir-Knudsen law). For modelling the combustion process, we employ the Eulerian Stochastic Field method and a laminar chemistry model. The simulations encompass two droplet diameter distributions, one with low and the other with significant diameter non-uniformity. Furthermore, nine distinct initial conditions are considered. Our findings indicate that adjusting the discretisation scheme or the evaporation model can yield simulation outcomes differing to a substantial degree, similar to the impact of modifying the actual properties of the spray droplets or their initial distribution. This is particularly pronounced in the case of sprays featuring a narrowed droplet diameter distribution. We found that the Abramzon–Sirignano model results in the most rapid autoignition. The TVD scheme contributes to a more even spatial distribution of the evaporated fuel, consequently facilitating quicker evaporation and faster autoignition. Using the WENO scheme generates higher fuel vapor gradients near the droplets, leading to less intense evaporation and slower fuel transport in space.

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.

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.

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.

Second-Order Accurate TVD Numerical Methods for Nonlocal Nonlinear Conservation Laws

We present a second-order accurate numerical method for a class of nonlocal nonlinear conservation laws called the "nonlocal pair-interaction model" which was recently introduced by Du, Huang, and LeFloch. Our numerical method uses second-order accurate reconstruction-based schemes for local conservation laws in conjunction with appropriate numerical integration. We show that the resulting method is total variation diminishing (TVD) and converges towards a weak solution. In fact, in contrast to local conservation laws, our second-order reconstruction-based method converges towards the unique entropy solution provided that the nonlocal interaction kernel satisfies a certain growth condition near zero. Furthermore, as the nonlocal horizon parameter in our method approaches zero we recover a well-known second-order method for local conservation laws. In addition, we answer several questions from the paper from Du, Huang, and LeFloch concerning regularity of solutions. In particular, we prove that any discontinuity present in a weak solution must be stationary and that, if the interaction kernel satisfies a certain growth condition, then weak solutions are unique. We present a series of numerical experiments in which we investigate the accuracy of our second-order scheme, demonstrate shock formation in the nonlocal pair-interaction model, and examine how the regularity of the solution depends on the choice of flux function.

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.

Numerical error control for second-order explicit TVD scheme with limiters in advection simulation

This paper analyzes the causes of the numerical errors in terms of numerical diffusion and compression arising from the use of explicit second-order total variation diminishing (TVD) schemes in one-dimensional advection simulation. It demonstrates that different TVD limiters may have very different performances in different advection simulations, because of the so-called numerical diffusion and compression. The accuracy of the computed results is found to depend on not only the limiter functions themselves but also the advection features such as the concentration distribution, advection velocity and time step, etc. According to such relations, the effective ranges of the MIN_MOD, Van Leer and SUPERBEE limiters are characterized by introducing a dimensionless parameter which reflects the key features of advections, aiming to provide an approach to select a proper TVD limiter in advection simulation.

Evolution of symmetric reconnection layer in the presence of parallel shear flow

The development of the structure of symmetric reconnection layer in the presence of a shear flow parallel to the antiparallel magnetic field component is studied by using a set of one-dimensional (1D) magnetohydrodynamic (MHD) equations. The Riemann problem is simulated through a second-order conservative TVD (total variation diminishing) scheme, in conjunction with Roe’s averages for the Riemann problem. The simulation results indicate that besides the MHD shocks and expansion waves, there exist some new small-scale structures in the reconnection layer. For the case of zero initial guide magnetic field (i.e., By0 = 0), a pair of intermediate shock and slow shock (SS) is formed in the presence of the parallel shear flow. The critical velocity of initial shear flow Vzc is just the Alfven velocity in the inflow region. As Vz∞ increases to the value larger than Vzc, a new slow expansion wave appears in the position of SS in the case Vz∞ < Vzc, and one of the current densities drops to zero. As plasma β increases, the out-flow region is widened. For By0 ≠ 0, a pair of SSs and an additional pair of time-dependent intermediate shocks (TDISs) are found to be present. Similar to the case of By0 = 0, there exists a critical velocity of initial shear flow Vzc. The value of Vzc is, however, smaller than the Alfven velocity of the inflow region. As plasma β increases, the velocities of SS and TDIS increase, and the out-flow region is widened. However, the velocity of downstream SS increases even faster, making the distance between SS and TDIS smaller. Consequently, the interaction between SS and TDIS in the case of high plasma β influences the property of direction rotation of magnetic field across TDIS. Thereby, a wedge in the hodogram of tangential magnetic field comes into being. When β → ∞, TDISs disappear and the guide magnetic field becomes constant.

Shock-capturing method using characteristic-based dissipation filters in pressure-based algorithm

In this paper, an efficient blending procedure based on the pressure-based algorithm is presented to solve the compressible Euler equations on a non-orthogonal mesh with collocated finite volume formulation. The boundedness criteria for this procedure are determined from total variation diminishing (TVD) schemes with and without applying of artificial compression method (ACM) of Harten as a control switch of dissipation. The fluxes of the convected quantities including mass flow rate are approximated by using the characteristic-based TVD and TVD/ACM methods. The algorithm is tested for steady-state inviscid flows at different Mach numbers ranging from the transonic to the supersonic regime and the results are compared with the existing numerical solutions. The comparisons show that the ACM is a useful technique to modify standard high-resolution schemes, which prevents the smearing of discontinuities and improves the resolution of shocks in the pressure-based algorithm. Aside from having the ability of accurately capturing shocked flows, this approach also accelerates the convergence rate of the solution in the supersonic flows with only a maximum increase of 5% in the operations with respect to standard second-order TVD schemes.

The Savage‐Hutter theory: A system of partial differential equations for avalanche flows of snow, debris, and mud

AbstractThe Savage‐Hutter (SH) equations of granular avalanche flows are a hyperbolic system of equations determining the distribution of depth and depth‐averaged velocity components tangential to the sliding bed. We review the equations and point out the geometrical complexities to which these equations have been generalized. Because of the hyperbolicity of the equations, successful numerical modelling is challenging, particularly when large gradients of the physical variables occur, e.g. for a moving front or possibly formed shock waves in avalanche flows if velocities change from supercritical to subcritical e.g. during the deposition. Numerical schemes solving these free surface flows must be able to cope with smooth as well as non‐smooth solutions. In this paper several numerical methods are applied to solve the SH equations and compared, including traditional difference schemes, e.g. central and upstream difference schemes, as well as high‐resolution NOC (Non‐Oscillatory Central Differencing) schemes, in which several second‐order TVD (Total Variation Diminishing) limiters and a third‐order ENO (Essentially Non‐Oscillatory) cell reconstruction scheme are used. Results show that the high‐resolution schemes, particularly the NOC scheme with the Minmod TVD limiter or the van Leer limiter, provide excellent performances. In the SH theory the material response is expressed by only two phenomenological parameters – the internal and the bed friction angles. Parameter investigations show that avalanche flows are much more sensitive against variations of the bed friction angle than that of the internal angle of friction. Effects due to a pressure dependence of the bed friction angle and lateral variations of the basal topography are therefore also numerically examined.

Numerical problems in semiconductor simulation using the hydrodynamic model: a second-order finite difference scheme

In this paper, a second-order Total Variation Diminishing (TVD) finite difference scheme of upwind type is employed for the numerical approximation of the classical hydrodynamic model for semiconductors proposed by Bløtekjær and Baccarani–Wordeman. In particular, the high-order hyperbolic fluxes are evaluated by a suitable extrapolation on adjacent cells of the first-order fluxes of Roe, while total variation diminishing is achieved by limiting the slopes of the discrete Riemann invariants using the so-called Flux Corrected Transport approach. Extensive numerical simulations are performed on a submicron n +− n− n + ballistic diode. The numerical experiments show that the spurious oscillations arising in the electron current are not completely suppressed by the TVD scheme, and can lead to serious numerical instabilities when the solution of the hydrodynamic model is non-smooth and the computational mesh is coarse. The accuracy of the numerical method is investigated in terms of conservation of the steady electron current. The obtained results show that the second-order scheme does not behave much better than a corresponding first-order one due to a poor performance of the slope limiters caused by the presence of local extrema of the Riemann invariant associated with the hyperbolic system.

Two-dimensional free surface flow in branch channels by a finite-volume TVD scheme

Free surface flow, in particular caused by dam-breaks in branch channels or other arbitrary geometrical rivers is an attention getting subject to the engineering practice, however the studies are few to be reported. In this paper a finite-volume total variation diminishing (TVD) scheme is presented for modeling unsteady free surface flows caused by dam-breaks in branch channels. In order to extend the finite-difference TVD scheme to finite-volume form, a mesh topology is defined relating a node and an element. The solver is implemented for the 2D shallow water equations on arbitrary quadrilateral meshes, and based upon a second-order hybrid TVD scheme with an optimum-selected limiter in the space discretization and a two-step Runge–Kutta approach in the time discretization. Verification for two typical dam-break problems is carried out by comparing the present results with others and very good agreement is obtained. The present algorithm is then used to predict the characteristics of free surface flows due to dam breaking in branch channels, for example, in a symmetrical trifurcated channel and a natural bifurcated channel, on coarse meshes and fine meshes, respectively. The characteristics of complex unsteady free surface flows in these examples are clearly shown.

Aerodynamic computations for a transonic projectile at angle of attack by total variation diminishing schemes

Several total variation diminishing (TVD) schemes and the Beam-Warming solution algorithm have been incorporated in a computer code for solving three-dimensional time-dependent , Reynold's averaged NavierStokes equations in generalized coordinates. A finite-volume fully implicit, approximately factored scheme is employed with a time-step convergence acceleration. Transonic turbulent flows past a scant ogive cylinder boat tail projectile at Mach numbers 0.94 and 0.97 including base flow region were studied. Conditions at 0- and 4-deg angle of attack were computed. Results of zero angle of attack showed that third-order TVD schemes gave little improvements over the second-order TVD schemes. Van Leer's third-order MUSCL-type TVD scheme yields somewhat better results and faster convergence than Chakravarthy's upwind-biased TVD scheme. The convergence rate of the Beam-Warming scheme is similar to that of the second-order TVD schemes. The Beam-Warming scheme gives comparative results over most of the projectile except regions of coarse grids because the scheme is very sensitive to grid resolution. For angle-of-attack computations, Van Leer's third-order MUSCL-type TVD scheme still gives better agreement with measurement.

Third-order nonoscillatory schemes for the Euler equations

Two time-level third-order finite-difference shock-capturin g schemes based on applying the characteristic flux difference splitting to a modified flux that may have high-order accuracy and either monotonicity preserving or essentially nonoscillatory (ENO) property have been developed for the Euler equations of gas dynamics. Two ways to achieve high-order accuracy are described. One is based on upstream interpolation using Lagrange's formula with van Leer's smoothness monitors. The other is based on ENO interpolation using reconstruction via primitive function approaches. For multidimensional problems, dimensional splitting is adopted for explicit schemes, and standard alternating direction implicit approximate factorization procedures are used for implicit schemes. Numerical examples to illustrate the performance of the proposed schemes are given. ERY recently, a new class of uniformly high-order accurate, essentially nonoscillatory (ENO) schemes has been developed by Harten and Osher,1 Marten,2 and Harten et al.3'4 They presented a hierarchy of uniformly high-order accurate schemes that generalize Godunov's scheme5 and its secondorder accurate MUSCL extension6'7 and total variation diminishing (TVD) schemes8'9 to an arbitrary order of accuracy. In contrast to the earlier second-order TVD schemes that drop to first-order accuracy at local extrema and maintain second-order accuracy in smooth regions, the new ENO schemes are uniformly high-order accurate throughout, even at critical points. Theoretical results for the scalar coefficient case and numerical results for the scalar conservation law and for the one-dimensional Euler equations of gas dynamics have been reported with highly accurate results. Preliminary results for two-dimensional problems were reported in Ref. 2. In this paper, following van Leer,10'11 Harten,2 and Harten et al.3'4 we describe a class of third-order, essentially nonoscillatory shock-capturing schemes for the Euler equations of gas dynamics. These schemes are obtained by applying the characteristic flux-difference splitting to an appropriately modified flux vector that may have high-order accuracy and nonoscillatory property. Third-order schemes are constructed using upstream interpolation and ENO interpolation. Both explicit and implicit schemes are derived. Implicit schemes for two-dimensional Euler equations in general coordinates are also given. We apply the resulting schemes to simulate one-dimensional and two-dimensional unsteady shock tube flows and steady two-dimensional flows involving strong shocks to illustrate the performance of the schemes. In Sec. II, characteristic properties of the Euler equations related to the numerical advections are briefly summarized. Upstream interpolation using Lagrange's formula to generate a class of two time-level, 2p + 1 space point, (2p - l)thorder accurate schemes is described in Sec. III. In particular, a third-order scheme with smoothness monitors due to van Leer10 is described. In Sec. IV, the essentially nonoscillatory interpolation of Harten2 and Harten et al.3'4 using reconstruction via primitive function approach is employed to yield a third-order nonoscillatory scheme.

Numerical investigation of separated flow around a bent-nose biconic

The purpose of this article is to investigate the separated flow around the bent-nose biconic, in which the second-order TVD (total variation diminishing) method is applied and the numerical flux is a Harten-Yee type. The governing equation is a thin layer Navier-Stokes equation. The boundary condition on the wall is the adiabatic wall. The whole flow field is laminar. The numerical conditions are as follows: Mach number 7.0, Reynolds number 8.38·10 5. The surface pressure distribution at the leeward side is a little higher in the vicinity of the symmetrical plane. This seems to be caused by the development of the vortex at the leeward side due to the separated flow. The location of a separation line and the size are in very good agreement with the experiment. Also the secondary separation is captured very clearly. The characteristics of the separated flow around the bent-nose biconic are clarified.