Smooth Extrema Research Articles

Positivity-preserving discontinuous spectral element methods for compressible multi-species flows

We introduce a novel positivity-preserving numerical stabilisation approach for high-order discontinuous spectral element approximations of compressible multi-species flows. The underlying stabilisation method is the adaptive entropy filtering approach (Dzanic and Witherden, J. Comput. Phys., 468, 2022), which is extended to the conservative formulation of the multi-species flow equations. We show that the straightforward enforcement of entropy constraints in the filter yields poor results around species interfaces and propose an adaptive switch for the entropy bounds based on the convergence properties of the pressure field which drastically improves its performance for multi-species flows. The proposed approach is shown in a variety of numerical experiments applied to the multi-species Euler and Navier–Stokes equations computed on unstructured grids, ranging from shock-fluid interaction problems to three-dimensional viscous flow instabilities. We demonstrate that the approach can retain the high-order accuracy of the underlying numerical scheme even at smooth extrema, ensure the positivity of the species density and pressure in the vicinity of shocks and contact discontinuities, and accurately predict small-scale flow features with minimal numerical dissipation.

Monotonicity-Preserving Lax–Wendroff Scheme for Solving Scalar Hyperbolic Conservation Laws

In this paper, we construct the monotonicity-preserving Lax–Wendroff (MP-Lax–Wendroff) scheme based on the MP scheme as proposed by Suresh and Huynh [11], which is a high-order and high-resolution method for hyperbolic conservation laws. It is well known that the total-variation-diminishing (TVD) methods possess the order reduction at smooth extremum points. However, the MP scheme not only preserves the non-oscillatory behavior of the procedure, but also prevents the order reduction of the method. We provide an analysis of the MP procedure using the numerical Lax–Wendroff flux sophistically in detail. Due to lack of robustness of the MP scheme, a rigorous and efficient MP-Lax–Wendroff scheme is introduced to enhance the robustness of the MP process. This means that a TVD numerical flux is applied to damp the oscillations at disturbed discontinuity points to get the high-resolution property. Therefore, we apply this idea to construct the improved MP-Lax–Wendroff scheme (we call it MP-R-Lax–Wendroff scheme). The computational performance of the MP-R-Lax–Wendroff scheme for linear and nonlinear hyperbolic conservation laws is conducted through some prototype examples which in turn validate our theoretical results.

A Characteristic-Featured Shock Wave Indicator for Conservation Laws Based on Training an Artificial Neuron

In this work, we use exact solutions of one-dimensional Burgers equation to train an artificial neuron as a shock wave detector. The expression of the artificial neuron detector is then modified into a practical form to reflect admissible jump of eigenvalues. We show the working mechanism of the practical form is consistent with compressing or intersecting of characteristic curves. In addition, we prove there is indeed a discontinuity inside the cell detected by the practical form, and smooth extrema and large gradient regions are never marked. As a result, we apply the practical form to numerical schemes as a shock wave indicator with its easy extension to multi-dimensional conservation laws. Numerical results are present to demonstrate the robustness of the present indicator under Runge–Kutta Discontinuous Galerkin framework, its performance is generally compared to TVB-based indicators more efficiently and accurately. To treat the initial inadmissible jumps, including linear contact discontinuities and those evolving into rarefaction waves, a preliminary strategy of combining a traditional indicator in the beginning with the present indicator is suggested. We believe the present indicator can be applied to unstructured mesh in the future.

Numerical simulation of a bulb turbine with an eddy-preserving limiter scheme

The eddy-preserving limiter scheme has been demonstrated to outperform the conventional van Albada limiter for Monotone Upstream-centered Schemes for Conservation Laws (MUSCL) for vortical flows. It reduces the dissipation by inactivating the conventional van Albada limiter in the interpolations of the velocity components on the swirl plane of the vortex. In this work, we extend the limiter for the interpolation of pressure, since a minimum pressure often exists along the axis of a free vortex. The conventional van Albada limiter is inactivated in the interpolation of pressure at smooth extrema. A 3D inviscid vortex advection case is employed to demonstrate the effects of the novel scheme. The extended eddy-preserving limiter scheme has been demonstrated to be able to further minimize the dissipation of the vortex than the original eddy-preserving limiter scheme. Finally, the scheme is applied to the BulbT test case and the numerical results are compared against experimental data.

Spatial Divergence of Multimode Acoustic Beams Formed by a Vertical Array in a Refracting Oceanic Waveguide

We consider the patterns manifested during the formation of the spatial intensity distribution of an acoustic field excited by a vertical array in a deep-water oceanic waveguide with an underwater sound channel open to the surface and an acoustically transparent bottom. The study focuses on the dependence of the spatial divergence of multimode beams on the angle of compensation of an array emitting a tone signal. It is established that if the size of the array aperture significantly exceeds the optimal size, then at a certain angle of compensation appreciably differing from zero, a multimode beam forms with minimal diffraction divergence even in the absence of bending points and smooth extrema of the dependence of the length of the ray cycle on the ray parameter.

Central WENO Schemes Through a Global Average Weight

A novel central weighted essentially non-oscillatory (central WENO; CWENO)-type scheme for the construction of high-resolution approximations to discontinuous solutions to hyperbolic systems of conservation laws is presented. This procedure is based on the construction of a global average weight using the whole set of Jiang–Shu smoothness indicators associated to every candidate stencil. By this device one does not to have to rely on ideal weights, which, under certain stencil arrangements and interpolating point locations, do not define a convex combination of the lower-degree interpolating polynomials of the corresponding sub-stencils. Moreover, this procedure also prevents some cases of accuracy loss near smooth extrema that are experienced by classical WENO and CWENO schemes. These properties result in a more flexible scheme that overcomes these issues, at the cost of only a few additional computations with respect to classical WENO schemes and with a smaller cost than classical CWENO schemes. Numerical examples illustrate that the proposed CWENO schemes outperform both the traditional WENO and the original CWENO schemes.

A single-stage flux-corrected transport algorithm for high-order finite-volume methods

We present a new limiter method for solving the advection equation using a high-order, finite-volume discretization. The limiter is based on the flux-corrected transport algorithm. We modify the classical algorithm by introducing a new computation for solution bounds at smooth extrema, as well as improving the pre-constraint on the high-order fluxes. We compute the high-order fluxes via a method of lines approach with fourth order Runge-Kutta as the time integrator. For computing low-order fluxes, we select the corner transport upwind method due to its improved stability over donor-cell upwind. Several spatial differencing schemes are investigated for the high-order flux computation, including centered difference and upwind schemes. We show that the upwind schemes perform well on account of the dissipation of high wavenumber components. The new limiter method retains high-order accuracy for smooth solutions and accurately captures fronts in discontinuous solutions. Further, we need only apply the limiter once per complete time step.

Flux-corrected transport algorithms for continuous Galerkin methods based on high order Bernstein finite elements

This work extends the flux-corrected transport (FCT) methodology to arbitrary order continuous finite element discretizations of scalar conservation laws on simplex meshes. Using Bernstein polynomials as local basis functions, we constrain the total variation of the numerical solution by imposing local discrete maximum principles on the Bézier net. The design of accuracy-preserving FCT schemes for high order Bernstein–Bézier finite elements requires the development of new algorithms and/or generalization of limiting techniques tailored for linear and multilinear Lagrange elements. In this paper, we propose (i) a new discrete upwinding strategy leading to local extremum bounded low order approximations with compact stencils, (ii) high order variational stabilization based on the difference between two gradient approximations, and (iii) new localized limiting techniques for antidiffusive element contributions. The optional use of a smoothness indicator, based on a second derivative test, makes it possible to potentially avoid unnecessary limiting at smooth extrema and achieve optimal convergence rates for problems with smooth solutions. The accuracy of the proposed schemes is assessed in numerical studies for the linear transport equation in 1D and 2D.

Determining the effective resolution of advection schemes. Part II: Numerical testing

Numerical models of fluid flows calculate the resolved flow at a given grid resolution. The smallest wave resolved by the numerical scheme is deemed the effective resolution. Advection schemes are an important part of the numerical models used for computational fluid dynamics. For example, in atmospheric dynamical cores they control the transport of tracers. For linear schemes solving the advection equation, the effective resolution can be calculated analytically using dispersion analysis. Here, a numerical test is developed that can calculate the effective resolution of any scheme (linear or non-linear) for the advection equation.The tests are focused on the use of non-linear limiters for advection schemes. It is found that the effective resolution of such non-linear schemes is very dependent on the number of time steps. Initially, schemes with limiters introduce large errors. Therefore, their effective resolution is poor over a small number of time steps. As the number of time steps increases the error of non-linear schemes grows at a smaller rate than that of the linear schemes which improves their effective resolution considerably. The tests highlight that a scheme that produces large errors over one time step might not produce a large accumulated error over a number of time steps. The results show that, in terms of effective-resolution, there is little benefit in using higher than third-order numerical accuracy with traditional limiters. The use of weighted essentially non-oscillatory (WENO) schemes, or relaxed and quasi-monotonic limiters, which allow smooth extrema, can eliminate this reduction in effective resolution and enable higher than third-order accuracy.

Fifth-Order Mapped Semi-Lagrangian Weighted Essentially Nonoscillatory Methods Near Certain Smooth Extrema

Fifth-order mapped semi-Lagrangian weighted essentially nonoscillatory (WENO) methods at certain smooth extrema are developed in this study. The schemes contain the mapped semi-Lagrangian finite volume (M-SL-FV) WENO 5 method and the mapped compact semi-Lagrangian finite difference (M-C-SL-FD) WENO 5 method. The weights in the more common scheme lose accuracy at certain smooth extrema. We introduce mapped weighting to handle the problem. In general, a cell average is applied to construct the M-SL-FV WENO 5 reconstruction, and the M-C-SL-FD WENO 5 interpolation scheme is proposed based on an interpolation approach. An accuracy test and numerical examples are used to demonstrate that the two schemes reduce the loss of accuracy and improve the ability to capture discontinuities.

A Solvable Chaotic Map of q-Phase Transitions Caused by Diverging Local Expansion Rates

In chaotic dynamical systems, non-hyperbolicity or bifurcations such as an onset of type-I intermittency (a tangent bifurcation) and a crisis, which can be observed, for example, at both ends of the period-three window of the logistic map, cause characteristic large fluctuations in a finite range of the local expansion rate, whose long time average is equal to the maximum Lyapunov exponent. This appears as a non-analytic rate function of the local expansion rate. Some researchers call it a ‘‘q-phase transition’’. 1) For a one-dimensional chaotic map xnþ1 ¼ f ðxnÞ, the local expansion rate � ðxnÞ is given by � ðxn Þ¼ log jdf ðxnÞ=dxnj .I f the map has a smooth extremum at x ¼ x� , � ðx� Þ goes to negative infinity. Although a smooth extremum is a special case of non-hyperbolicity in a one-dimensional map, diverging local expansion rates due to non-hyperbolicity in general chaotic dynamical systems such as higher-dimensional maps and differential equations also cause a nonanalytic rate function of the local expansion rate. 1) The Lorenz plot with the classical parameters 2) has a cuspshaped maximum, at which the local expansion rate goes to

Weights Design For Maximal Order WENO Schemes

Weighted essentially non-oscillatory (WENO) finite difference schemes, developed by Liu et al. (Comput Phys 115(1):200---212, 1994) and improved by Jiang and Shu (Comput Phys 126(1):202---228, 1996), are one of the most popular methods to approximate the solutions of hyperbolic equations. But these schemes fail to provide maximal order accuracy near smooth extrema, where the first derivative of the solution becomes zero. Some authors have addressed this problem with different weight designs. In this paper we focus on the weights proposed by Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009). They propose new weights to provide faster weight convergence than those presented in Borges et al. (J Comput Phys 227:3191---3211, 2008) and deduce some constraints on the weights parameters to guarantee that the WENO scheme has maximal order for sufficiently smooth solutions with an arbitrary number of vanishing derivatives. We analyze the scheme with the weights proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009) and prove that near discontinuities it achieves worse orders than classical WENO schemes. In order to solve these accuracy problems, we define new weights, based on those proposed in Yamaleev and Carpenter (J Comput Phys 228:4248---4272, 2009), and get some constraints on the weights parameters to guarantee maximal order accuracy for the resulting schemes.

High-order solution-adaptive central essentially non-oscillatory (CENO) method for viscous flows

A high-order, central, essentially non-oscillatory (CENO), finite-volume scheme in combination with a block-based adaptive mesh refinement (AMR) algorithm is proposed for solution of the Navier–Stokes equations on body-fitted multi-block mesh. In contrast to other ENO schemes which require reconstruction on multiple stencils, the proposed CENO method uses a hybrid reconstruction approach based on a fixed central stencil. This feature is crucial to avoiding the complexities associated with multiple stencils of ENO schemes, providing high-order accuracy at relatively lower computational cost as well as being very well suited for extension to unstructured meshes. The spatial discretization of the inviscid (hyperbolic) fluxes combines an unlimited high-order k-exact least-squares reconstruction technique following from the optimal central stencil with a monotonicity-preserving, limited, linear, reconstruction algorithm. This hybrid reconstruction procedure retains the unlimited high-order k-exact reconstruction for cells in which the solution is fully resolved and reverts to the limited lower-order counterpart for cells with under-resolved/discontinuous solution content. Switching in the hybrid procedure is determined by a smoothness indicator. The high-order viscous (elliptic) fluxes are computed to the same order of accuracy as the hyperbolic fluxes based on a k-order accurate cell interface gradient derived from the unlimited, cell-centred, reconstruction. A somewhat novel h-refinement criterion based on the solution smoothness indicator is used to direct the steady and unsteady mesh adaptation. The proposed numerical procedure is thoroughly analyzed for advection–diffusion problems characterized by the full range of Péclet numbers, and its predictive capabilities are also demonstrated for several inviscid and laminar flows. The ability of the scheme to accurately represent solutions with smooth extrema and yet robustly handle under-resolved and/or non-smooth solution content (i.e., shocks and other discontinuities) is shown. Moreover, the ability to perform mesh refinement in regions of under-resolved and/or non-smooth solution content is also demonstrated.

Positivity Preservation and Advection Algorithms with Applications to Edge Plasma Turbulence

The unique conditions of edge tokamak plasmas motivate the employment of efficient numerical methods that can robustly handle steep temperature and density gradients. To that end, we compare an Arakawa finite difference technique with some recent high-resolution upwind methods that are designed to minimize nonmonotonic overshoots while preserving the accuracy of solutions at smooth extrema. Versions of these algorithms are able to rigorously preserve positivity when that is physically relevant (such as for the advection of particle density or temperature). We explain in detail the use of these methods for the nonlinear Poisson bracket, an operator applicable to neutral fluid, gyrofluid, gyrokinetic, and general Hamiltonian simulations. For one-dimensional passive advection, the high-resolution upwind techniques maintain monotonicity and approach minimal levels of phase error and dissipation, especially at long wavelengths. In a two-dimensional incompressible Navier--Stokes vortex merging problem we find that extrema-preserving methods can resolve details at lower resolution than the Arakawa technique, are less dissipative than traditional finite volume methods while still minimizing overshoots and ensuring positivity, and model nonlinear cascade behavior fairly well without additional subgrid damping.

A parameter-free smoothness indicator for high-resolution finite element schemes

AbstractThis paper presents a postprocessing technique for estimating the local regularity of numerical solutions in high-resolution finite element schemes. A derivative of degree p ≥ 0 is considered to be smooth if a discontinuous linear reconstruction does not create new maxima or minima. The intended use of this criterion is the identification of smooth cells in the context of p-adaptation or selective flux limiting. As a model problem, we consider a 2D convection equation discretized with bilinear finite elements. The discrete maximum principle is enforced using a linearized flux-corrected transport algorithm. The deactivation of the flux limiter in regions of high regularity makes it possible to avoid the peak clipping effect at smooth extrema without generating spurious undershoots or overshoots elsewhere.

Slope limiting for discontinuous Galerkin approximations with a possibly non‐orthogonal Taylor basis

SUMMARYThe use of high‐order polynomials in discontinuous Galerkin (DG) approximations to convection‐dominated transport problems tends to cause a violation of the maximum principle in regions where the derivatives of the solution are large. In this paper, we express the DG solution in terms of Taylor basis functions associated with the cell average and derivatives at the center of the cell. To control the (derivatives of the) discontinuous solution, the values at the vertices of each element are required to be bounded by the means. This constraint is enforced using a hierarchical vertex‐based slope limiter to constrain the coefficients of the Taylor polynomial in a conservative manner starting with the highest‐order terms. The loss of accuracy at smooth extrema is avoided by taking the maximum of the correction factors for derivatives of order p and higher. No free parameters, oscillation detectors, or troubled cell markers are involved. In the case of a non‐orthogonal Taylor basis, the same limiter is applied to the vector of discretized time derivatives before the multiplication by the off‐diagonal part of the consistent mass matrix. This strategy leads to a remarkable gain of accuracy, especially in the case of simplex meshes. A numerical study is performed for a 2D convection equation discretized with linear and quadratic finite elements. Copyright © 2012 John Wiley & Sons, Ltd.

An Improved Method for Phase Extraction Based on Gray Extremum

Surface deformation of an object can be measured by digital shearography through non-contact measurement with a simple device. It is always difficult to remove the noise of the speckle interference. Consequently extracting phase from a single fringe pattern without carrier-frequency is complex. Based on gray extremum, an improved method for phase extraction is proposed by using the improved image binarization algorithm in this paper. Through comparing the average value of eight neighbourhood points around each pixel with threshold value, the quality of binary image can be improved, and then the skeleton line can be extracted. Finally a smooth extremum image can be obtained. Experimental results show that the proposed method is simple, convenient and reliable.

Dynamic programming in the presence of indeterminacy

480 Indeterminacy must be addressed in optimization in many scientific and technical fields, such as the reg ulation of production processes and the creation of new materials and technologies. The required out comes are optimal parameter values, optimal trajecto ries toward the goal, and often models of the process (response surfaces). Traditional dynamic program ming permits the derivation of optimal trajectories, but only where a model of the process exists—that is, only in conditions of complete determinacy. Optimi zation methods of search type permit the determina tion of optimal parameter values but do not provide optimal trajectories for their attainment. Such meth ods do not provide means of deriving a model of the process. To derive optimal parameter values, optimal trajec tories, and a model of the process, we propose integra tion of dynamic programming, evolutionary planning (search based optimization), and adaptive identifica tion. The proposed method is based on a successive iterative approach: formulation of experiments within a small region of the response surface; construction of a mathematical model in that region; extrapolation of the model to the whole response surface; identifica tion of the optimal trajectory on the extrapolated sur face; determination of the trajectory within a small specified vicinity; subsequent repetition of these oper ations for a new region of the response surface in the direction of motion of the optimal trajectory; and con tinuation until the optimal trajectories and optimal parameters have been established. So as to be specific, we consider examples in which the response surface is a smooth single extremum sur face (Fig. 1a) or has a more complex configuration with gullies (Fig. 1b). First we select the coordinates of the initial point A0 in Fig. 2. Around this point, we construct a regular simplex and conduct experiments to determine the values of the response surface function. From the results, by the least squares method, we construct a model of the form y = a1x1 + a2x2 + a0. The model is extended over the whole region of investigation. Then, by dynamic programming, we determine the optimal trajectory from point A1 to point A0 in the plane y I = ⎯1.9x1 – 1.9x2 + 1.8. The model is extended to a distance specified by the simplex increment (points 2 and 3 in Fig. 3a). The resulting trajectory is fixed in region I (seg ment A0–A2 in Fig. 2). Around point A2, which is the final point of the trajectory, we again construct a regu Dynamic Programming in the Presence of Indeterminacy

Weakly divergent ray and diffraction beams in oceanic waveguides

Conditions that should be satisfied by the sound velocity profile of an oceanic waveguide for the dependence of the ray cycle length on the ray phase velocity to contain smooth extrema are formulated. The extrema correspond to weakly divergent ray beams forming “caustic” beams. It is found that diffraction effects cause a considerable smoothing of the sharp extrema that occur in the dependence of the interference period of neighboring modes on their phase velocity. As a result, in addition to the weakly divergent ray beams, weakly divergent diffraction beams and the corresponding “diffraction” caustics can be formed.

A Genuinely High Order Total Variation Diminishing Scheme for One-Dimensional Scalar Conservation Laws

It is well known that finite difference or finite volume total variation diminishing (TVD) schemes solving one-dimensional scalar conservation laws degenerate to first order accuracy at smooth extrema [S. Osher and S. Chakravarthy, SIAM J. Numer. Anal., 21 (1984), pp. 955–984], thus TVD schemes are at most second order accurate in the $L^1$ norm for general smooth and nonmonotone solutions. However, Sanders [Math. Comp., 51 (1988), pp. 535–558] introduced a third order accurate finite volume scheme which is TVD, where the total variation is defined by measuring the variation of the reconstructed polynomials rather than the traditional way of measuring the variation of the grid values. By adopting the definition of the total variation for the numerical solutions as in [R. Sanders, Math. Comp., 51 (1988), pp. 535–558], it is possible to design genuinely high order accurate TVD schemes. In this paper, we construct a finite volume scheme which is TVD in this sense with high order accuracy (up to sixth order) in the $L^1$ norm. Numerical tests for a fifth order accurate TVD scheme will be reported, which include test cases from traffic flow models.