Efficient algorithms for polynomial interpolation and numerical differentiation
Algorithms based on Newton formula for polynomial interpolation and numerical differentiation
Krogh, F. T.↗
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Matrix-based, finite-difference algorithms for computational acoustics
A compact numerical algorithm is introduced for simulating multidimensional acoustic waves. The algorithm is expressed in terms of a set of matrix coefficients on a three-point spatial grid that approximates the acoustic wave equation with a discretization error of O(h exp 5). The method is based on tracking a local phase variable and its implementation suggests a convenient coordinate splitting along with natural intermediate boundary conditions. Results are presented for oblique plane waves and compared with other procedures. Preliminary computations of acoustic diffraction are also considered.
Davis, Sanford↗
A test suite for magnetohydrodynamical simulations
A collection is presented of MHD problems that will provide researchers who have interests in modeling MHD phenomena with a battery of tests (a 'test suite') for calibrating their numerical algorithms. Use of these tests will provide a common reference for comparison of different numerical MHD algorithms. The test suite includes both 1D and 2D problems. Taken together, these problems test the abilities of a numerical method to propagate accurately all of the MHD wave families in moving and stationary media, to capture MHD shocks and contact discontinuities, and to model accurately Lorentz force terms in multiple dimensions. An example solution of each test problem is presented. Diagnostic convergence-testing procedures that provide a quantitative evaluation of MHD algorithms are demonstrated.
Stone, James M.↗
Verification of Numerical Programs: From Real Numbers to Floating Point Numbers
Numerical algorithms lie at the heart of many safety-critical aerospace systems. The complexity and hybrid nature of these systems often requires the use of interactive theorem provers to verify that these algorithms are logically correct. Usually, proofs involving numerical computations are conducted in the infinitely precise realm of the field of real numbers. However, numerical computations in these algorithms are often implemented using floating point numbers. The use of a finite representation of real numbers introduces uncertainties as to whether the properties veri ed in the theoretical setting hold in practice. This short paper describes work in progress aimed at addressing these concerns. Given a formally proven algorithm, written in the Program Verification System (PVS), the Frama-C suite of tools is used to identify sufficient conditions and verify that under such conditions the rounding errors arising in a C implementation of the algorithm do not affect its correctness. The technique is illustrated using an algorithm for detecting loss of separation among aircraft.
Goodloe, Alwyn E.↗
Calculation of two-dimensional inlet flow fields in a supersonic free stream by an implicit marching code with nonorthogonal mesh generation: User's manual
An implicit, shock-capturing finite-difference code which is used to calculate two-dimensional inlet flow fields in a supersonic free stream is explained. The Euler equations are subjected to general nonorthogonal transformation and a body-fitted coordinate system is employed. The mathematical formulation of the problem is given along with the numerical algorithm. Initial and boundary conditions, numerical stability, program limitations, and accuracy is discussed. An overall program logic as well as instructions for program use and operation are also furnished.
Biringen, S. H.↗
Comparative Study on High-Order Positivity-preserving WENO Schemes
In gas dynamics and magnetohydrodynamics flows, physically, the density and the pressure p should both be positive. In a standard conservative numerical scheme, however, the computed internal energy is obtained by subtracting the kinetic energy from the total energy, resulting in a computed p that may be negative. Examples are problems in which the dominant energy is kinetic. Negative may often emerge in computing blast waves. In such situations the computed eigenvalues of the Jacobian will become imaginary. Consequently, the initial value problem for the linearized system will be ill posed. This explains why failure of preserving positivity of density or pressure may cause blow-ups of the numerical algorithm. The adhoc methods in numerical strategy which modify the computed negative density and/or the computed negative pressure to be positive are neither a conservative cure nor a stable solution. Conservative positivity-preserving schemes are more appropriate for such flow problems. The ideas of Zhang & Shu (2012) and Hu et al. (2012) precisely address the aforementioned issue. Zhang & Shu constructed a new conservative positivity-preserving procedure to preserve positive density and pressure for high-order WENO schemes by the Lax-Friedrichs flux (WENO/LLF). In general, WENO/LLF is too dissipative for flows such as turbulence with strong shocks computed in direct numerical simulations (DNS) and large eddy simulations (LES). The new conservative positivity-preserving procedure proposed in Hu et al. (2012) can be used with any high-order shock-capturing scheme, including high-order WENO schemes using the Roe's flux (WENO/Roe). The goal of this study is to compare the results obtained by non-positivity-preserving methods with the recently developed positivity-preserving schemes for representative test cases. In particular the more difficult 3D Noh and Sedov problems are considered. These test cases are chosen because of the negative pressure/density most often exhibited by standard high-order shock-capturing schemes. The simulation of a hypersonic nonequilibrium viscous shock tube that is related to the NASA Electric Arc Shock Tube (EAST) is also included. EAST is a high-temperature and high Mach number viscous nonequilibrium flow consisting of 13 species. In addition, as most common shock-capturing schemes have been developed for problems without source terms, when applied to problems with nonlinear and/or sti source terms these methods can result in spurious solutions, even when solving a conservative system of equations with a conservative scheme. This kind of behavior can be observed even for a scalar case (LeVeque & Yee 1990) as well as for the case consisting of two species and one reaction (Wang et al. 2012). For further information concerning this issue see (LeVeque & Yee 1990; Griffiths et al. 1992; Lafon & Yee 1996; Yee et al. 2012). This EAST example indicated that standard high-order shock-capturing methods exhibit instability of density/pressure in addition to grid-dependent discontinuity locations with insufficient grid points. The evaluation of these test cases is based on the stability of the numerical schemes together with the accuracy of the obtained solutions.
WENO↗
AutoBayes Program Synthesis System System Internals
This lecture combines the theoretical background of schema based program synthesis with the hands-on study of a powerful, open-source program synthesis system (Auto-Bayes). Schema-based program synthesis is a popular approach toward program synthesis. The lecture will provide an introduction into this topic and discuss how this technology can be used to generate customized algorithms. The synthesis of advanced numerical algorithms requires the availability of a powerful symbolic (algebra) system. Its task is to symbolically solve equations, simplify expressions, or to symbolically calculate derivatives (among others) such that the synthesized algorithms become as efficient as possible. We will discuss the use and importance of the symbolic system for synthesis. Any synthesis system is a large and complex piece of code. In this lecture, we will study Autobayes in detail. AutoBayes has been developed at NASA Ames and has been made open source. It takes a compact statistical specification and generates a customized data analysis algorithm (in C/C++) from it. AutoBayes is written in SWI Prolog and many concepts from rewriting, logic, functional, and symbolic programming. We will discuss the system architecture, the schema libary and the extensive support infra-structure. Practical hands-on experiments and exercises will enable the student to get insight into a realistic program synthesis system and provides knowledge to use, modify, and extend Autobayes.
Statistical Algorithms↗
Comparitive Study of High-Order Positivity-Preserving WENO Schemes
In gas dynamics and magnetohydrodynamics flows, physically, the density ρ and the pressure p should both be positive. In a standard conservative numerical scheme, however, the computed internal energy is The ideas of Zhang & Shu (2012) and Hu et al. (2012) precisely address the aforementioned issue. Zhang & Shu constructed a new conservative positivity-preserving procedure to preserve positive density and pressure for high-order Weighted Essentially Non-Oscillatory (WENO) schemes by the Lax-Friedrichs flux (WENO/LLF). In general, WENO/LLF is obtained by subtracting the kinetic energy from the total energy, resulting in a computed p that may be negative. Examples are problems in which the dominant energy is kinetic. Negative ρ may often emerge in computing blast waves. In such situations the computed eigenvalues of the Jacobian will become imaginary. Consequently, the initial value problem for the linearized system will be ill posed. This explains why failure of preserving positivity of density or pressure may cause blow-ups of the numerical algorithm. The adhoc methods in numerical strategy which modify the computed negative density and/or the computed negative pressure to be positive are neither a conservative cure nor a stable solution. Conservative positivity-preserving schemes are more appropriate for such flow problems. too dissipative for flows such as turbulence with strong shocks computed in direct numerical simulations (DNS) and large eddy simulations (LES). The new conservative positivity-preserving procedure proposed in Hu et al. (2012) can be used with any high-order shock-capturing scheme, including high-order WENO schemes using the Roe's flux (WENO/Roe). The goal of this study is to compare the results obtained by non-positivity-preserving methods with the recently developed positivity-preserving schemes for representative test cases. In particular the more di cult 3D Noh and Sedov problems are considered. These test cases are chosen because of the negative pressure/density most often exhibited by standard high-order shock-capturing schemes. The simulation of a hypersonic nonequilibrium viscous shock tube that is related to the NASA Electric Arc Shock Tube (EAST) is also included. EAST is a high-temperature and high Mach number viscous nonequilibrium ow consisting of 13 species. In addition, as most common shock-capturing schemes have been developed for problems without source terms, when applied to problems with nonlinear and/or sti source terms these methods can result in spurious solutions, even when solving a conservative system of equations with a conservative scheme. This kind of behavior can be observed even for a scalar case as well as for the case consisting of two species and one reaction.. This EAST example indicated that standard high-order shock-capturing methods exhibit instability of density/pressure in addition to grid-dependent discontinuity locations with insufficient grid points. The evaluation of these test cases is based on the stability of the numerical schemes together with the accuracy of the obtained solutions.
Schemes↗
Sparsified time-dependent Fourier neural operators for fusion simulations
This paper presents a sparsified Fourier neural operator for coupled time-dependent partial differential equations (ST-FNO) as an efficient machine learning surrogate for fluid and particle-based fusion codes such as NIMROD (Non-Ideal Magnetohydrodynamics with Rotation - Open Discussion) and GTC (Gyrokinetic Toroidal Code). ST-FNO leverages the structures in the governing equations and utilizes neural operators to represent Green's function-like numerical operators in the corresponding numerical solvers. Once trained, ST-FNO can rapidly and accurately predict dynamics in fusion devices compared with first-principle numerical algorithms. In general, ST-FNO represents an efficient and accurate machine learning surrogate for numerical simulators for multi-variable nonlinear time-dependent partial differential equations, with the proposed architectures and loss functions. The efficacy of ST-FNO has been demonstrated using quiescent H-mode simulation data from NIMROD and kink-mode simulation data from GTC. The ST-FNO H-mode results show orders of magnitude reduction in memory and central processing unit usage in comparison with the numerical solvers in NIMROD when computing fields over a selected poloidal plane. The ST-FNO kink-mode results achieve a factor of 2 reduction in the number of parameters compared to baseline FNO models without accuracy loss.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Bayesian optimization algorithms for accelerator physics
Accelerator physics relies on numerical algorithms to solve optimization problems in online accelerator control and tasks such as experimental design and model calibration in simulations. The effectiveness of optimization algorithms in discovering ideal solutions for complex challenges with limited resources often determines the problem complexity these methods can address. The accelerator physics community has recognized the advantages of Bayesian optimization algorithms, which leverage statistical surrogate models of objective functions to effectively address complex optimization challenges, especially in the presence of noise during accelerator operation and in resource-intensive physics simulations. In this review article, we offer a conceptual overview of applying Bayesian optimization techniques toward solving optimization problems in accelerator physics. We begin by providing a straightforward explanation of the essential components that make up Bayesian optimization techniques. We then give an overview of current and previous work applying and modifying these techniques to solve accelerator physics challenges. Finally, we explore practical implementation strategies for Bayesian optimization algorithms to maximize their performance, enabling users to effectively address complex optimization challenges in real-time beam control and accelerator design. Published by the American Physical Society 2024
43 PARTICLE ACCELERATORS↗
Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations
New numerical algorithms are devised (PSC algorithms) for following fronts propagating with curvature-dependent speed. The speed may be an arbitrary function of curvature, and the front can also be passively advected by an underlying flow. These algorithms approximate the equations of motion, which resemble Hamilton-Jacobi equations with parabolic right-hand-sides, by using techniques from the hyperbolic conservation laws. Non-oscillatory schemes of various orders of accuracy are used to solve the equations, providing methods that accurately capture the formation of sharp gradients and cusps in the moving fronts. The algorithms handle topological merging and breaking naturally, work in any number of space dimensions, and do not require that the moving surface be written as a function. The methods can be used also for more general Hamilton-Jacobi-type problems. The algorithms are demonstrated by computing the solution to a variety of surface motion problems.
Osher, Stanley↗
Transformation of two and three-dimensional regions by elliptic systems
Finite difference methods for composite grids were analyzed. It was observed that linear interpolation between grids would suffice only where low order accuracy was required. In the context of fluid flow, this would be in regions where the flow was essentially free stream. Higher order interpolation schemes were also investigated. The well known quadratic and cubic interpolating polynomials would increase the formal accuracy of the overall numerical algorithm. However, it can also be shown that the stability of the algorithm may be adversely affected. Further numerical results are needed in order to assess the nature of this instability induced by the interpolation procedure. Finally, error analysis and the order of difference expressions on general curvilinear coordinates are discussed.
Thompson, J. F.↗
Computation of multi-dimensional viscous supersonic jet flow
A new method has been developed for two- and three-dimensional computations of viscous supersonic flows with embedded subsonic regions adjacent to solid boundaries. The approach employs a reduced form of the Navier-Stokes equations which allows solution as an initial-boundary value problem in space, using an efficient noniterative forward marching algorithm. Numerical instability associated with forward marching algorithms for flows with embedded subsonic regions is avoided by approximation of the reduced form of the Navier-Stokes equations in the subsonic regions of the boundary layers. Supersonic and subsonic portions of the flow field are simultaneously calculated by a consistently split linearized block implicit computational algorithm. The results of computations for a series of test cases relevant to internal supersonic flow is presented and compared with data. Comparison between data and computation are in general excellent thus indicating that the computational technique has great promise as a tool for calculating supersonic flow with embedded subsonic regions. Finally, a User's Manual is presented for the computer code used to perform the calculations.
Kim, Y. N.↗
Computation of multi-dimensional viscous supersonic flow
A method has been developed for two- and three-dimensional computations of viscous supersonic jet flows interacting with an external flow. The approach employs a reduced form of the Navier-Stokes equations which allows solution as an initial-boundary value problem in space, using an efficient noniterative forward marching algorithm. Numerical instability associated with forward marching algorithms for flows with embedded subsonic regions is avoided by approximation of the reduced form of the Navier-Stokes equations in the subsonic regions of the boundary layers. Supersonic and subsonic portions of the flow field are simultaneously calculated by a consistently split linearized block implicit computational algorithm. The results of computations for a series of test cases associated with supersonic jet flow is presented and compared with other calculations for axisymmetric cases. Demonstration calculations indicate that the computational technique has great promise as a tool for calculating a wide range of supersonic flow problems including jet flow. Finally, a User's Manual is presented for the computer code used to perform the calculations.
Buggeln, R. C.↗
A numerical study of transient heat and mass transfer in crystal growth
A numerical analysis of transient heat and solute transport across a rectangular cavity is performed. Five nonlinear partial differential equations which govern the conservation of mass, momentum, energy and solute concentration related to crystal growth in solution, are simultaneously integrated by a numerical method based on the SIMPLE algorithm. Numerical results showed that the flow, temperature and solute fields are dependent on thermal and solutal Grashoff number, Prandtl number, Schmidt number and aspect ratio. The average Nusselt and Sherwood numbers evaluated at the center of the cavity decrease markedly when the solutal buoyancy force acts in the opposite direction to the thermal buoyancy force. When the solutal and thermal buoyancy forces act in the same direction, however, Sherwood number increases significantly and yet Nusselt number decreases. Overall effects of convection on the crystal growth are seen to be an enhancement of growth rate as expected but with highly nonuniform spatial growth variations.
Han, Samuel Bang-Moo↗
Derivative interface conditions for multiblock grids
Methods are developed for computing numerical solutions along block boundaries, even when there is a discontinuity in the grid lines or slopes. The technique is based on matching derivatives and does not require overlapping and interpolation of solution values at block boundaries. The comparison of block boundary values is implicit and has proven to be stable for both implicit and explicit numerical algorithms. Examples are included for the numerical solution of the Euler equations for compressible flow on grids with both grid line discontinuities and discontinuous slopes at block boundaries.
Mastin, C. W.↗
Direct numerical simulations of on-demand vortex generators: Mathematical formulation
The objective of the present research is the development and application of efficient adaptive numerical algorithms for the study, via direct numerical simulations, of active vortex generators. We are using innovative computational schemes to investigate flows past complex configurations undergoing arbitrary motions. Some of the questions we try to answer are: Can and how may we control the dynamics of the wake? What is the importance of body shape and motion in the active control of the flow? What is the effect of three-dimensionality in laboratory experiments? We are interested not only in coupling our results to ongoing, related experimental work, but furthermore to develop an extensive database relating the above mechanisms to the vortical wake structures with the long-range objective of developing feedback control mechanisms. This technology is very important to aircraft, ship, automotive, and other industries that require predictive capability for fluid mechanical problems. The results would have an impact in high angle of attack aerodynamics and help design ways to improve the efficiency of ships and submarines (maneuverability, vortex induced vibration, and noise).
Koumoutsakos, Petros↗
Achieving Higher Order Accuracy in Space in Hydrodynamic Simulations of Self-Gravitating Gas
Modern astrophysical simulation codes employ a variety of numerical algorithms capable of achieving higher-order accuracy in both space and time. Albeit they succeed in achieving an effective higher spatial resolution and in suppressing the numerical damping of waves, to our knowledge, all current astrophysical simulations invoking self-gravity are limited to second-order accuracy in space. If we can devise an algorithm to evaluate self-gravity with a higher-order spatial accuracy, we can better the evaluation of the gravitational acceleration and gravitational energy release which dictate the evolution of many astrophysical systems. Herein, we present a numerical algorithm for self-gravitating hydrodynamics capable of achieving fourth order accuracy for a given density distribution on a Cartesian uniform grid. First, we derive the cell-averaged gravitational potential at fourth-order accuracy from the cell-averaged density by solving the Poisson equation. Next, we obtain the cell average of the product of the density and gravitational acceleration, which differs from the cell-averaged density multiplied by the cell-averaged gravitational acceleration. We then show the verification of the algorithm by applying it to critical test problems: (1) maintaining equilibria of self-gravitating slabs, even upon advection, (2) evolving a polytropic sphere with a massive power-law envelope, and (3) conservation of specific entropy during the propagation of a sound wave.
79 ASTRONOMY AND ASTROPHYSICS↗