Lowest Order Solution Research Articles

Accelerating high order discontinuous Galerkin solvers using neural networks: Wall bounded flows

High order solvers are accurate but computationally expensive as they require small time steps to advance the solution in time. In this work we include a corrective forcing to a low order solution to improve the accuracy while advancing in time with larger time steps, and achieve fast computations. The work uses a discontinuous Galerkin framework, where the polynomial order, inside each mesh element, can be varied to provide low or high accuracy. The corrective forcing is included for each high order Gauss nodal point in the mesh. This work is a continuation of [1, 2], where we extend the methodology to wall bounded flows. Namely, we adapt the methodology to a turbulent channel at Reτ = 182. In this case, we use three neural networks to correct different regions of the flow, which are distinguished by their y+ distance to the wall. The methodology is able to correct the low resolution simulation to attain flow statistics that are comparable to high order simulations. We include comparisons for the mean, Reynolds stresses and shear stress on the wall. We achieve good predictions using the corrected low order solution, in mean velocity and its corresponded fluctuations, as well as the shear stress on the wall.

The Hybrid Pathway to Flexible Power Turbines, Part II: Fast Data Transfer Methods Between Varying Fidelity Simulations, to Enable Efficient Conjugate Thermal Field Prediction

Abstract The global drive toward renewable energy is imposing challenging operating requirements on power turbines. Flexible load-leveling applications must accept more frequent and demanding start-stop cycles. Full transient analyses are too computationally expensive for real-time simulation across all operating regimes so monitoring relies on sparse physical measurements. Alone, these sparse data lack the fidelity for real-time prediction of a complex thermal field. A novel hybrid methodology is proposed, coupling data across a range of fidelities to bridge the limitations in the individual analyses. Combining several fidelity methods in parallel; low-order models, corrected by real-time physical measurements, are calibrated with high-fidelity simulations. A newly developed low-order thermal network code is used to predict the thermal field in real-time. High-fidelity flow characteristics are routinely transferred to the decoupled low-order solution. A critical enabling feature of this hybrid approach is the fast data interpolation between differing fidelity numerical simulations. This paper evaluates a spatial Kriging method for robust data transfer between two different fidelity mesh, tested in the case of thermal profile prediction of a power turbine. Additionally, a novel coordinate-based hash mapping process is demonstrated for the fast high-to-low fidelity data transfer. Localized hashing allows independent, parallel, nearest neighbor search at significantly reduced computational cost. The demonstrated method facilitates fast mesh pairing, necessary to support the real-time hybrid method for thermal field prediction during turbine transient operation.

The symmetric breathers and lumps of the Boussinesq equation using the Alice–Bob transformation and Hirota’s bilinear derivative method

Many two-place physical problems can be explicitly presented as related events model named Alice–Bob systems. In this paper, an integrable Alice–Bob Boussinesq system is introduced via the Boussinesq equation with parameters, which may meet the symmetry transformation of Psˆx (parity with a shift) and Tdˆt (time reversal with a delay). After constructing an Bäcklund transformation, the system has rich symmetry solutions with the aid of auxiliary functions. The structures of obtained soliton solutions, such as the breathers, lumps and their hybrids, are all satisfied the Pˆsx or Tˆdt symmetry. To illustrate the symmetric characteristic, some lower-order solutions and the related dynamic structures are explicitly presented. The residual symmetry and its finite transformation for this system are also verified.

Electron heating in kinetic-Alfvén-wave turbulence.

We report analytical and numerical investigations of subion-scale turbulence in low-beta plasmas using a rigorous reduced kinetic model. We show that efficient electron heating occurs and is primarily due to Landau damping of kinetic Alfvén waves, as opposed to Ohmic dissipation. This collisionless damping is facilitated by the local weakening of advective nonlinearities and the ensuing unimpeded phase mixing near intermittent current sheets, where free energy concentrates. The linearly damped energy of electromagnetic fluctuations at each scale explains the steepening of their energy spectrum with respect to a fluid model where such damping is excluded (i.e., a model that imposes an isothermal electron closure). The use of a Hermite polynomial representation to express the velocity-space dependence of the electron distribution function enables us to obtain an analytical, lowest-order solution for the Hermite moments of the distribution, which is borne out by numerical simulations.

A high-order deferred correction method for the solution of free boundary problems using penalty iteration, with an application to American option pricing

This paper presents a high-order deferred correction algorithm combined with penalty iteration for solving free and moving boundary problems, using a fourth-order finite difference method. Typically, when free boundary problems are solved on a fixed computational grid, the order of the solution is low due to the discontinuity in the solution at the free boundary, even if a high-order method is used. Using a detailed error analysis, we observe that the order of convergence of the solution can be increased to fourth-order by solving successively corrected finite difference systems, where the corrections are derived from the previously computed lower order solutions. The corrections are applied solely to the right-hand side, and leave the finite difference matrix unchanged. The penalty iterations converge quickly given a good initial guess. We demonstrate the accuracy and efficiency of our algorithm using several examples. Numerical results show that our algorithm gives fourth-order convergence for both the solution and the free boundary location. We also test our algorithm on the challenging American put option pricing problem. Our algorithm gives the expected high-order convergence.

Novel Wentzel–Kramers–Brillouin Solutions to the Nonisentropic Helmholtz Equation in a Nonuniform Duct With Mean Temperature Gradient and Mean Flow

Abstract We derive the generalized Helmholtz equation (GHE) governing nonisentropic acoustic fluctuations in a quasi 1D duct with nonuniform cross section, mean temperature gradient, and nonuniform mean flow. Nonisentropic effects are included via heat conduction terms in the mean and fluctuating energy equations. To derive the Helmholtz equation exclusively in terms of the fluctuating pressure field, a relationship between density and pressure fluctuations is needed, which is shown to be a second-order differential equation for nonisentropic motions. Novel analytical solutions that are accurate for both low/high frequencies and small/large mean gradients are presented for the GHE based on the Wentzel–Kramers–Brillouin (WKB) method. WKB solutions are developed using the ansatz that the pressure fluctuation field has a travelling wave form, p^(x)=exp[∫0x(a+ib)dx], where x is the axial coordinate. Substituting this form into the Helmholtz equation yields coupled, nonlinear ordinary differential equations (ODEs) for a and b. Analytical solutions to the ODEs are obtained using the approximations of high frequency and slowly varying mean properties. This simplification allows us to obtain the lower order solutions b0 and a0. We then enhance solution accuracy by using a0 to solve for b1 without any approximations. Finally, b1 is employed to get a1, giving us the higher order solution. The p^ calculated from (a1, b1) is in good to excellent agreement with numerical solution of the GHE for both low and high frequencies and for a range of mean Mach numbers, including M¯≳1.

From P/E Ratio to Fuzzy Infinite Spreadsheet—Mathematically Rigorous Derivations of the Zeroth and the First Order Solutions of Rate of Return

The P/E Ratio (Price/Earning) is one of the most popular concepts in stock analysis, yet its exact interpretation is lacking. Most stock investors know the P/E Ratio as a financial indicator with the useful characteristics of being relatively time-invariant. In this paper, a rigorous mathematical derivation of the P/E Ratio is presented. The derivation shows that, in addition to its assumptions, the P/E Ratio can be considered the zeroth order solution to the rate of return on investment. The commonly used concept of the Capitalization Rate (Cap Rate = Net Income / Price) in real estate investment analysis can also be similarly derived as the zeroth order solution of the rate of return on real estate investment. This paper also derives the first order solution to the rate of return (Return = Dividend/Price + Growth) with its assumptions. Both the zeroth and the first order solutions are derived from the exact future accounting equation (Cash Return = Sum of Cash Flow + Cash from Resale). The exact equation has been used in the derivation of the exact solution of the rate of return. Empirically, as an illustration of an actual case, the rates of return are 3%, 73%, and 115% for a stock with 70% growth rate for, respectively, the zeroth order, the first order, and the exact solution to the rate of return; the stock doubled its price in 2004. This paper concludes that the zero-th, the first order, and the exact solution of the rate of return all can be derived mathematically from the same exact equation, which, thus, forms a rigorous mathematical foundation for investment analysis, and that the low order solutions have the very practical use in providing the analytically calculated initial conditions for the iterative numerical calculation for the exact solution. The solution of value belongs to recently classified Culture Level Quotient CLQ = 10 and is in the process of being updated by fuzzy logic with its range of tolerance for predicting market crashes to advance to CLQ = 2.

Accelerating high order discontinuous Galerkin solvers using neural networks: 1D Burgers’ equation

High order discontinuous Galerkin methods allow accurate solutions through the use of high order polynomials inside each mesh element. Increasing the polynomial order leads to high accuracy, but increases the cost. On the one hand, high order polynomials require more restrictive time steps when using explicit temporal schemes, and on the other hand the quadrature rules lead to more costly evaluations per iteration. We propose to accelerate high order discontinuous Galerkin methods using Neural Networks. To this aim, we train a Neural Network using a high order discretisation, to extract a corrective forcing that can be applied to a low order solution with the aim of recovering high order accuracy. With this corrective forcing term, we can run a low order solution (low cost) and correct the solution to obtain high order accuracy. We provide error bounds to quantify the various errors included in the methodology (e.g. related to the discretisation or the Neural Network) . The methodology and bounds are examined for a variety of meshes, polynomial orders and viscosity values for the 1D viscous Burgers’ equation. The result show good accuracy and accelerations specially when considering high polynomial orders.

A high-order/low-order (HOLO) algorithm for preserving conservation in time-dependent low-rank transport calculations

Dynamical low-rank (DLR) approximation methods have previously been developed for time-dependent radiation transport problems. One crucial drawback of DLR is that it does not conserve important quantities of the calculation, which limits the applicability of the method. Here we address this conservation issue by solving a low-order equation with closure terms computed via a high-order solution calculated with DLR. We observe that the high-order solution well approximates the closure term, and the low-order solution can be used to correct the conservation bias in the DLR evolution. We also apply the linear discontinuous Galerkin method for the spatial discretization. We then demonstrate with the numerical results that this so-called high-order/low-order (HOLO) algorithm is conservative without sacrificing computational efficiency and accuracy.

Stress-driven spin-down of a viscous fluid within a spherical shell

Abstract

Degenerate solutions for the spatial discrete Hirota equation

In this paper, some degenerate solutions of the spatial discrete Hirota equation are constructed via the degenerate idea of positon solution. Under the zero seed solution, the n-positon is obtained by N-fold degenerate Darboux transformation (DT). The degenerate DT is taking the degenerate limit $$\lambda _{j}\rightarrow \lambda _{1}$$ for the eigenvalues $$\lambda _{j}(j=1,2, 3, \ldots , N)$$ of N-fold DT and then performing the high-order Taylor expansion near $$\lambda _{1}$$ . Considering the universal Darboux transformation, breather is obtained from the nonzero seed. Then, a new type of breather solution can be produced by using the same degenerated method and higher-order Taylor expansion for eigenvalues in determinant expression of breather solution. The explicit determinants of breather-positon solution and positon solution are constructed, respectively, and the complicated and significant dynamics of low-order solution are also revealed.

Multilevel quasidiffusion method with mixed-order time discretization for multigroup thermal radiative transfer problems

In this paper we present a numerical method for solving multigroup thermal radiative transfer (TRT) problems in 2D Cartesian geometry. It is based on the Quasidiffusion (aka Variable Eddington Factor) method and defined by the multilevel system of multigroup high-order radiative transfer (RT) equations and multigroup and grey low-order equations for moments of the intensity with the exact closures. We apply time integration schemes of different orders of accuracy to approximate the high-order and low-order equations. The first-order scheme is used for the high-order RT equations. The second-order scheme is applied to the low-order equations. This improves the accuracy of the TRT solution while using robust and relatively inexpensive scheme for the high-order RT equations. The solution of the low-order equations is non-monotonic because the hyperbolic low-order Quasidiffusion (QD) equations are discretized by the second-order scheme. To reduce non-monotonicity of the low-order solution we apply a monotonization procedure to the discretized time-dependent low-order equations based on the Limited-Trapezoidal method. Numerical results of the Fleck-Cummings test are presented to demonstrate performance of the developed mixed-order time integration scheme for the multilevel system of high-order and low-order QD equations. We use this TRT test problem to analyze the convergence in time of the mixed-order scheme.

Near-axis expansion of stellarator equilibrium at arbitrary order in the distance to the axis

A direct construction of equilibrium magnetic fields with toroidal topology at arbitrary order in the distance from the magnetic axis is carried out, yielding an analytical framework able to explore the landscape of possible magnetic flux surfaces in the vicinity of the axis. This framework can provide meaningful analytical insight into the character of high-aspect-ratio stellarator shapes, such as the dependence of the rotational transform and the plasma beta limit on geometrical properties of the resulting flux surfaces. The approach developed here is based on an asymptotic expansion on the inverse aspect ratio of the ideal magnetohydrodynamics equation. The analysis is simplified by using an orthogonal coordinate system relative to the Frenet–Serret frame at the magnetic axis. The magnetic field vector, the toroidal magnetic flux, the current density, the field line label and the rotational transform are derived at arbitrary order in the expansion parameter. Moreover, a comparison with a near-axis expansion formalism employing an inverse coordinate method based on Boozer coordinates (the so-called Garren–Boozer construction) is made, where both methods are shown to agree at lowest order. Finally, as a practical example, a numerical solution using a W7-X equilibrium is presented, and a comparison between the lowest-order solution and the W7-X magnetic field is performed.

Efficient Convergence for a Higher-Order Unstructured Finite Volume Solver for Compressible Flows

This Paper presents a three-dimensional higher-order-accurate finite volume algorithm for the solution of steady-state compressible flow problems. Higher-order accuracy is achieved by constructing a piecewise continuous representation of the average solution values using the -exact reconstruction scheme. The pseudo-transient continuation method is employed to reduce the solution of the discretized system of nonlinear equations into the solution of a series of linear systems, which are subsequently solved using the generalized minimal residual (GMRES) method. This Paper considers several preconditioning methods in conjunction with different matrix reordering algorithms and shows that the proposed preconditioner based on inner GMRES iterations can enhance the convergence speed and reduce the memory cost of the solver. Moreover, when starting from a lower-order solution as the initial condition, this Paper shows that ramping up the Courant–Friedrichs–Lewy (CFL) number accelerates the convergence rate. Finally, this Paper verifies the developed finite volume algorithm by solving a set of test problems, in which optimal solution convergence with mesh refinement is attained.

Monolithic convex limiting for continuous finite element discretizations of hyperbolic conservation laws

Using the theoretical framework of algebraic flux correction and invariant domain preserving schemes, we introduce a monolithic approach to convex limiting in continuous finite element schemes for linear advection equations, nonlinear scalar conservation laws, and hyperbolic systems. In contrast to flux-corrected transport (FCT) algorithms that apply limited antidiffusive corrections to bound-preserving low-order solutions, our new limiting strategy exploits the fact that these solutions can be expressed as convex combinations of bar states belonging to a convex invariant set of physically admissible solutions. Each antidiffusive flux is limited in a way which guarantees that the associated bar state remains in the invariant set and preserves appropriate local bounds. There is no free parameter and no need for limit fluxes associated with the consistent mass matrix of time derivative term separately. Moreover, the steady-state limit of the nonlinear discrete problem is well defined and independent of the pseudo-time step. In the case study for the Euler equations, the components of the bar states are constrained sequentially to satisfy local maximum principles for the density, velocity, and specific total energy in addition to positivity preservation for the density and pressure. The results of numerical experiments for standard test problems illustrate the ability of built-in convex limiters to resolve steep fronts in a sharp and nonoscillatory manner.

An adaptive domain-based POD/ECM hyper-reduced modeling framework without offline training

This work presents a reduced-order modeling framework that precludes the need for offline training and adaptively adjusts its lower-order solution space as the analysis progresses. The analysis starts with a fully-solved step and elements are clustered based on their strain response. Elements with the highest strains are solved with a local/global approach in which degrees of freedom from elements undergoing the highest amount of nonlinearity are fully-solved and the rest is approximated by a Proper Orthogonal Decomposition (POD) reduced model with full integration. Elements belonging to the remaining clusters are subjected to a hyper-reduction step using the Empirical Cubature Method (ECM). Online error estimators are used to trigger a retraining process once the reduced solution space becomes inadequate. The performance of the framework is assessed through a series of numerical examples featuring a material model with pressure-dependent plasticity.

Bright and dark rogue internal waves: The Gardner equation approach.

We have found "bright and dark" solutions of the Gardner equation which can model internal rogue waves in three-layer fluids. We provide the first four "bright" and "dark" exact rational solutions to the Gardner equation. These are the lowest-order solutions of the corresponding hierarchies of rogue-wave solutions of this equation. They have been obtained from the rogue-wave solutions of a modified Korteweg-de Vries equation by using a Lorentz-type transformation. The maximal (and minimal) amplitudes and the background levels of these solutions for arbitrary order are deduced, based on the lowest-order examples. These solutions can be useful for explanations of extremely large amplitude internal waves in the ocean, as well as for abnormally large-amplitude waves in other areas of nonlinear physics, such as optics and dusty plasmas.

A Weighted Least-Squares Transport Equation Compatible with Source Iteration and Voids

Second-order forms of the transport equation allow the use of continuous finite elements (CFEMs). This can be desired in multiphysics calculations where other physics require CFEM discretizations. Second-order transport operators are generally self-adjoint, yielding symmetric positive-definite (SPD) matrices, which allow the use of efficient linear algebra solvers with an enormous advantage in memory usage.Least-squares (LS) forms of the transport equation can circumvent the void problems of other second-order forms but are almost always nonconservative. Additionally, the standard LS form is not compatible with discrete ordinates method (SN) iterative solution techniques such as source iteration. A new form of the LS transport equation has recently been developed that is compatible with voids and standard SN iterative solution techniques. Performing nonlinear diffusion acceleration (NDA) using an independently differenced low-order equation enforces conservation for the whole system and makes this equation suitable for reactor physics calculations. In this context, “independent” means that both the transport and low-order solutions converge to the same scalar flux and current as the spatial mesh is refined, but for a given mesh, the solutions are not necessarily equal.In this paper we show that introducing a weight function into this LS equation improves issues with causality and can render our equation equal to the self-adjoint angular flux (SAAF) equation. Causality is a principle of the transport equation that states that information travels only downstream along characteristics. This principle can be violated numerically. We show how to limit the weight function in voids and demonstrate the effect of this limit on accuracy. Using the C5G7 benchmark, we compare our method to the SAAF formulation with a void treatment (SAAFτ) that is not self-adjoint and has a nonsymmetric coefficient matrix. We show that the weighted LS equation with NDA gives acceptable accuracy relative to the SAAFτ equation while maintaining a SPD system matrix.

Two-Level Transport Methods with Independent Discretization

In this paper, we present two-level nonlinear iteration methods for the transport equation in 1D slab geometry approximated by means of the linear discontinuous finite element method (LDFEM). We develop transport schemes based on the quasidiffusion (QD) method in which the low-order QD (LOQD) equations are discretized by the linear continuous finite element method (LCFEM). This requires a mapping of the LCFEM low-order solution to the LDFEM high-order solution to define the scattering term. Several mappings are proposed and analyzed. Another proposed transport discretization scheme is based on the step characteristics for the transport equation and LCFEM for the LOQD equations. We also develop new nonlinear synthetic acceleration (NSA) methods based on the LCFEM discretization of the QD equation. To gain iterative stability, the NSA algorithms are combined with the nonlinear Krylov acceleration method. We present numerical results that demonstrate performance and basic properties of the proposed discretization schemes and iterative solution methods.

Fundamental solutions to the regularised 13-moment equations: efficient computation of three-dimensional kinetic effects

Fundamental solutions (Green’s functions) are derived for the regularised 13-moment system (R13) of rarefied gas dynamics, for small departures from equilibrium; these solutions show the presence of Knudsen layers, associated with exponential decay terms, that do not feature in the solution of lower-order systems (e.g. the Navier–Stokes–Fourier equations). Incorporation of these new fundamental solutions into a numerical framework based on the method of fundamental solutions (MFS) allows for efficient computation of three-dimensional gas microflows at remarkably low computational cost. The R13-MFS approach accurately recovers analytic solutions for low-speed flow around a stationary sphere and heat transfer from a hot sphere (for which a new analytic solution has been derived), capturing non-equilibrium flow phenomena missing from lower-order solutions. To demonstrate the potential of the new approach, the influence of kinetic effects on the hydrodynamic interaction between approaching solid microparticles is calculated. Finally, a programme of future work based on the initial steps taken in this article is outlined.