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DOE OSTI2025

Explicit block encodings of boundary value problems for many-body elliptic operators

Simulation of physical systems is one of the most promising use cases of future digital quantum computers. In this work we systematically analyze the quantum circuit complexities of block encoding the discretized elliptic operators that arise extensively in numerical simulations for partial differential equations, including high-dimensional instances for many-body simulations. When restricted to rectangular domains with separable boundary conditions, we provide explicit circuits to block encode the many-body Laplacian with separable periodic, Dirichlet, Neumann, and Robin boundary conditions, using standard discretization techniques from low-order finite difference methods. To obtain high-precision, we introduce a scheme based on periodic extensions to solve Dirichlet and Neumann boundary value problems using a high-order finite difference method, with only a constant increase in total circuit depth and subnormalization factor. We then present a scheme to implement block encodings of differential operators acting on more arbitrary domains, inspired by Cartesian immersed boundary methods. We then block encode the many-body convective operator, which describes interacting particles experiencing a force generated by a pair-wise potential given as an inverse power law of the interparticle distance. This work provides concrete recipes that are readily translated into quantum circuits, with depth logarithmic in the total Hilbert space dimension, that block encode operators arising broadly in applications involving the quantum simulation of quantum and classical many-body mechanics.

Kharazi, Tyler [University of California, Berkeley↗

DOE OSTI2025

Toward Higher-order Accuracy in Self-gravitating Hydrodynamics

High-order algorithms have emerged in numerical astrophysics as a promising avenue to reduce truncation error (proportional to a power of the linear resolution Δ x ) with only a moderate increase to computational expense. Significant effort has been placed in the development of finite-volume algorithms for (magneto)hydrodynamics; however, state-of-the-art astrophysical simulations tightly couple a plenitude of physics, additionally including gravity, photon transport, cosmic-ray transport, chemistry, and/or diffusion, to name a few. Algorithms frequently operator-split this additional physics (often a first-order error in time) and/or adopt a model wherein their evaluation is limited to second-order accuracy in space. In this work, we present a fourth-order-accurate finite-volume scheme for self-gravitating hydrodynamics on a uniform Cartesian grid. The method supplies source terms for the gravitational acceleration ( ρ g ) and gravitational energy release ( ρ v · g ) associated with fourth-order-accurate solutions to the Poisson equation. Our scheme (1) guarantees the conservation of total linear momentum while (2) decreasing (in proportion to Δ x 4 ) the effects of spurious heating and/or cooling associated with truncation error in the gravity. We demonstrate expected convergence rates for the algorithm by measuring errors in test problems evolving self-gravity modified linear waves and 3D polytropic equilibria. We test robustness of the algorithm by integrating an induced “inside-out” adiabatic collapse. We also discuss a method to smoothly downgrade the solution to second-order spatial accuracy to avoid spurious overshoots near steep density and/or pressure gradients.

79 ASTRONOMY AND ASTROPHYSICS↗

NASA NTRS1975

High-order numerical solutions using cubic splines

The cubic spline collocation procedure for the numerical solution of partial differential equations was reformulated so that the accuracy of the second-derivative approximation is improved and parallels that previously obtained for lower derivative terms. The final result is a numerical procedure having overall third-order accuracy for a nonuniform mesh and overall fourth-order accuracy for a uniform mesh. Application of the technique was made to the Burger's equation, to the flow around a linear corner, to the potential flow over a circular cylinder, and to boundary layer problems. The results confirmed the higher-order accuracy of the spline method and suggest that accurate solutions for more practical flow problems can be obtained with relatively coarse nonuniform meshes.

Rubin, S. G.↗

NASA NTRS1986

Minimum-fuel control of high-order systems

The minimum-fuel control problem is of special interest in various space systems. To date, solutions of minimum-fuel control problems have been carried out for relatively low-order systems. Space structures, however, are generally characterized by a large number of degrees of freedom, so that minimum-fuel control of such systems requires a new approach. In the independent modal-space control (IMSC) method, the control laws are designed in the modal space for each mode independently. The minimum-fuel problem reduces to that of a set of independent second-order systems, so that minimum-fuel control is possible. This paper shows how the IMSC method can be used to control a space structure with a minimum amount of fuel. A numerical example is presented.

Shenhar, J.↗

NASA NTRS1989

A time accurate finite volume high resolution scheme for three dimensional Navier-Stokes equations

A time accurate, three-dimensional, finite volume, high resolution scheme for solving the compressible full Navier-Stokes equations is presented. The present derivation is based on the upwind split formulas, specifically with the application of Roe's (1981) flux difference splitting. A high-order accurate (up to the third order) upwind interpolation formula for the inviscid terms is derived to account for nonuniform meshes. For the viscous terms, discretizations consistent with the finite volume concept are described. A variant of second-order time accurate method is proposed that utilizes identical procedures in both the predictor and corrector steps. Avoiding the definition of midpoint gives a consistent and easy procedure, in the framework of finite volume discretization, for treating viscous transport terms in the curvilinear coordinates. For the boundary cells, a new treatment is introduced that not only avoids the use of 'ghost cells' and the associated problems, but also satisfies the tangency conditions exactly and allows easy definition of viscous transport terms at the first interface next to the boundary cells. Numerical tests of steady and unsteady high speed flows show that the present scheme gives accurate solutions.

Liou, Meng-Sing↗

NASA NTRS1989

A high-order Lagrangian-decoupling method for the incompressible Navier-Stokes equations

A high-order Lagrangian-decoupling method is presented for the unsteady convection-diffusion and incompressible Navier-Stokes equations. The method is based upon: (1) Lagrangian variational forms that reduce the convection-diffusion equation to a symmetric initial value problem; (2) implicit high-order backward-differentiation finite-difference schemes for integration along characteristics; (3) finite element or spectral element spatial discretizations; and (4) mesh-invariance procedures and high-order explicit time-stepping schemes for deducing function values at convected space-time points. The method improves upon previous finite element characteristic methods through the systematic and efficient extension to high order accuracy, and the introduction of a simple structure-preserving characteristic-foot calculation procedure which is readily implemented on modern architectures. The new method is significantly more efficient than explicit-convection schemes for the Navier-Stokes equations due to the decoupling of the convection and Stokes operators and the attendant increase in temporal stability. Numerous numerical examples are given for the convection-diffusion and Navier-Stokes equations for the particular case of a spectral element spatial discretization.

Ho, Lee-Wing↗

NASA NTRS1990

Counterrotating prop-fan simulations which feature a relative-motion multiblock grid decomposition enabling arbitrary time-steps

Improvements are presented of a computer algorithm developed for the time-accurate flow analysis of rotating machines. The flow model is a finite volume method utilizing a high-resolution approximate Riemann solver for interface flux definitions. The numerical scheme is a block LU implicit iterative-refinement method which possesses apparent unconditional stability. Multiblock composite gridding is used to orderly partition the field into a specified arrangement of blocks exhibiting varying degrees of similarity. Block-block relative motion is achieved using local grid distortion to reduce grid skewness and accommodate arbitrary time step selection. A general high-order numerical scheme is applied to satisfy the geometric conservation law. An even-blade-count counterrotating unducted fan configuration is chosen for a computational study comparing solutions resulting from altering parameters such as time step size and iteration count. The solutions are compared with measured data.

Janus, J. Mark↗

NASA NTRS1989

Understanding transition and turbulence through direct simulations

Direct simulations consist in solving the full Navier-Stokes equations, without any turbulence model, and describing all the detailed features of the flow. Usually the flows are three-dimensional and time-dependent and contain both coarse and fine structures, which makes the numerical task very challenging in terms of both the algorithm and the computational effort. Most of the work until now has involved spectral methods, which are highly accurate but not very flexible in terms of geometry or complex equations. For that reason, future work will also rely on high-order finite-difference or other methods. Direct simulations complement experimental work, and both contribute to the theory and the empirical knowledge of turbulence. Once such a simulation has been shown to be accurate, the flow field is completely known in three dimensions and time, including the pressure, the vorticity and any other quantity. On the other hand, most simulations to date solved the incompressible equations in rather simple geometries, and direct simulations will always be limited to moderate Reynolds numbers. Extensive simulations have been conducted in homogeneous turbulence, channel flows, boundary layers, and mixing layers. Much effort is devoted to addressing flows with compressibility and chemical reactions, and to new geometries such as a backward-facing step.

Spalart, P. R.↗

NASA NTRS1991

High-order ENO schemes applied to two- and three-dimensional compressible flow

High order essentially non-oscillatory (ENO) finite difference schemes are applied to the 2-D and 3-D compressible Euler and Navier-Stokes equations. Practical issues, such as vectorization, efficiency of coding, cost comparison with other numerical methods, and accuracy degeneracy effects, are discussed. Numerical examples are provided which are representative of computational problems of current interest in transition and turbulence physics. These require both nonoscillatory shock capturing and high resolution for detailed structures in the smooth regions and demonstrate the advantage of ENO schemes.

Shu, Chi-Wang↗

NASA NTRS1992

Unsteady flowfield simulation of ducted prop-fan configurations

A technique for the simulation of unsteady flows in and around complex rotating machinery is presented. Additional domain decomposition mechanisms are introduced which extend the range of applicability of software developed for the time-accurate simulation of rotating machinery flowfields. The flow models uses the unsteady 3D Euler equations, discretized as a finite-volume method, utilizing a high-resolution approximate Riemann solver for cell interface flux definitions. Multiblock domain decomposition is used to partition the field radially, axially, as well as circumferentially into an ordered arrangement of blocks which exhibit varying degrees of similarity. A general high-order numerical scheme is applied to satisfy the geometric conservation law. Two configurations are presented - ducted single rotation prop-fan and a rotor-deswirl vane combination which form a single stage fan. Comparisons are made to other numerical solutions for these geometries and to available experimental data.

Janus, J. M.↗

NASA NTRS1993

Numerical Simulation of a High Mach Number Jet Flow

The recent efforts to develop accurate numerical schemes for transition and turbulent flows are motivated, among other factors, by the need for accurate prediction of flow noise. The success of developing high speed civil transport plane (HSCT) is contingent upon our understanding and suppression of the jet exhaust noise. The radiated sound can be directly obtained by solving the full (time-dependent) compressible Navier-Stokes equations. However, this requires computational storage that is beyond currently available machines. This difficulty can be overcome by limiting the solution domain to the near field where the jet is nonlinear and then use acoustic analogy (e.g., Lighthill) to relate the far-field noise to the near-field sources. The later requires obtaining the time-dependent flow field. The other difficulty in aeroacoustics computations is that at high Reynolds numbers the turbulent flow has a large range of scales. Direct numerical simulations (DNS) cannot obtain all the scales of motion at high Reynolds number of technological interest. However, it is believed that the large scale structure is more efficient than the small-scale structure in radiating noise. Thus, one can model the small scales and calculate the acoustically active scales. The large scale structure in the noise-producing initial region of the jet can be viewed as a wavelike nature, the net radiated sound is the net cancellation after integration over space. As such, aeroacoustics computations are highly sensitive to errors in computing the sound sources. It is therefore essential to use a high-order numerical scheme to predict the flow field. The present paper presents the first step in a ongoing effort to predict jet noise. The emphasis here is in accurate prediction of the unsteady flow field. We solve the full time-dependent Navier-Stokes equations by a high order finite difference method. Time accurate spatial simulations of both plane and axisymmetric jet are presented. Jet Mach numbers of 1.5 and 2.1 are considered. Reynolds number in the simulations was about a million. Our numerical model is based on the 2-4 scheme by Gottlieb & Turkel. Bayliss et al. applied the 2-4 scheme in boundary layer computations. This scheme was also used by Ragab and Sheen to study the nonlinear development of supersonic instability waves in a mixing layer. In this study, we present two dimensional direct simulation results for both plane and axisymmetric jets. These results are compared with linear theory predictions. These computations were made for near nozzle exit region and velocity in spanwise/azimuthal direction was assumed to be zero.

Hayder, M. Ehtesham↗

NASA NTRS1993

Direct calculations of waves in fluid flows using a high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expressions for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations, and the numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the CFL limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about 6 to 8 grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 to 300 time steps. Three numerical examples of waves in compressible flow are included.

Yu, Sheng-Tao↗

NASA NTRS1993

Uniform high-order spectral methods for one- and two-dimensional Euler equations

Uniform high order spectral methods to solve multi-dimensional Euler equations for gas dynamics are discussed. Uniform high order spectral approximations with spectral accuracy in smooth regions of solutions are constructed by introducing the idea of the Essentially Non-Oscillatory (ENO) polynomial interpolations into the spectral methods. The authors present numerical results for the inviscid Burgers' equation, and for the one-dimensional Euler equations including the interactions between a shock wave and density disturbance, Sod's and Lax's shock tube problems, and the blast wave problem. The interaction between a Mach 3 two-dimensional shock wave and a rotating vortex is simulated.

Cai, Wei↗

NASA NTRS1994

An HP Adaptive Discontinuous Galerkin Method for Hyperbolic Conservation Laws

This dissertation addresses various issues for model classes of hyperbolic conservation laws. The basic approach developed in this work employs a new family of adaptive, hp-version, finite element methods based on a special discontinuous Galerkin formulation for hyperbolic problems. The discontinuous Galerkin formulation admits high-order local approximations on domains of quite general geometry, while providing a natural framework for finite element approximations and for theoretical developments. The use of hp-versions of the finite element method makes possible exponentially convergent schemes with very high accuracies in certain cases; the use of adaptive hp-schemes allows h-refinement in regions of low regularity and p-enrichment to deliver high accuracy, while keeping problem sizes manageable and dramatically smaller than many conventional approaches. The use of discontinuous Galerkin methods is uncommon in applications, but the methods rest on a reasonable mathematical basis for low-order cases and has local approximation features that can be exploited to produce very efficient schemes, especially in a parallel, multiprocessor environment. The place of this work is to first and primarily focus on a model class of linear hyperbolic conservation laws for which concrete mathematical results, methodologies, error estimates, convergence criteria, and parallel adaptive strategies can be developed, and to then briefly explore some extensions to more general cases. Next, we provide preliminaries to the study and a review of some aspects of the theory of hyperbolic conservation laws. We also provide a review of relevant literature on this subject and on the numerical analysis of these types of problems.

Bey, Kim S.↗

NASA NTRS1994

High-order centered difference methods with sharp shock resolution

In this paper we consider high-order centered finite difference approximations of hyperbolic conservation laws. We propose different ways of adding artificial viscosity to obtain sharp shock resolution. For the Riemann problem we give simple explicit formulas for obtaining stationary one and two-point shocks. This can be done for any order of accuracy. It is shown that the addition of artificial viscosity is equivalent to ensuring the Lax k-shock condition. We also show numerical experiments that verify the theoretical results.

Gustafsson, Bertil↗

NASA NTRS1994

SCISEAL: A CFD code for analysis of fluid dynamic forces in seals

A viewgraph presentation is made of the objectives, capabilities, and test results of the computer code SCISEAL. Currently, the seal code has: a finite volume, pressure-based integration scheme; colocated variables with strong conservation approach; high-order spatial differencing, up to third-order; up to second-order temporal differencing; a comprehensive set of boundary conditions; a variety of turbulence models and surface roughness treatment; moving grid formulation for arbitrary rotor whirl; rotor dynamic coefficients calculated by the circular whirl and numerical shaker methods; and small perturbation capabilities to handle centered and eccentric seals.

Athavale, Mahesh↗

NASA NTRS1994

Low-dissipation and -dispersion Runge-Kutta schemes for computational acoustics

In this paper, we investigate accurate and efficient time advancing methods for computational acoustics, where non-dissipative and non-dispersive properties are of critical importance. Our analysis pertains to the application of Runge-Kutta methods to high-order finite difference discretization. In many CFD applications multi-stage Runge-Kutta schemes have often been favored for their low storage requirements and relatively large stability limits. For computing acoustic waves, however, the stability consideration alone is not sufficient, since the Runge-Kutta schemes entail both dissipation and dispersion errors. The time step is now limited by the tolerable dissipation and dispersion errors in the computation. In the present paper, it is shown that if the traditional Runge-Kutta schemes are used for time advancing in acoustic problems, time steps greatly smaller than that allowed by the stability limit are necessary. Low-Dissipation and -Dispersion Runge-Kutta (LDDRE) schemes are proposed, based on an optimization that minimizes the dissipation and dispersion errors for wave propagation. Order optimizations of both single-step and two-step alternating schemes are considered. The proposed LDDRK schemes are remarkably more efficient than the classical Runge-Kutta schemes for acoustic computations. Moreover, low storage implementations of the optimized schemes are discussed. Special issues of implementing numerical boundary conditions in the LDDRK schemes are also addressed.

Hu, F. Q.↗

NASA NTRS1994

Direct calculations of waves in fluid flows using high-order compact difference scheme

The solution of the unsteady Euler equations by a sixth-order compact difference scheme combined with a fourth-order Runge-Kutta method is investigated. Closed-form expression for the amplification factors and their corresponding dispersion correlations are obtained by Fourier analysis of the fully discretized, two-dimensional Euler equations. The numerical dissipation, dispersion, and anisotropic effects are assessed. It is found that the Courant-Friedrichs-Lewy (CFL) limit for stable calculations is about 0.8. For a CFL number equal to 0.6, the smallest wavelength which is resolved without numerical damping is about six - eight grid nodes. For phase speeds corresponding to acoustic waves, the corresponding time period is resolved by about 200 - 300 time steps. Three numerical examples of waves in compressible flow are included: (1) sound propagation in a duct with linear shear, (2) linear wave growth in a compressible free shear layer, and (3) vortex pairing in a compressible free shear layer perturbed at two frequencies.

Yu, Sheng-Tao↗