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Results for “numerical differentiation”
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Data-driven linear time advance operators for the acceleration of plasma physics simulation
In this study, we demonstrate the application of data-driven linear operator construction for time advance with a goal of accelerating plasma physics simulation. We apply dynamic mode decomposition (DMD) to data produced by the nonlinear SOLPS-ITER (Scrape-off Layer Plasma Simulator - International Thermonuclear Experimental Reactor) plasma boundary code suite in order to estimate a series of linear operators and monitor their predictive accuracy via online error analysis. We find that this approach defines when these dynamics can be represented by a sequence of approximate linear operators and is essential for providing consistent projections when compared to an unconstrained application. For linear diffusion and advection–diffusion fluid test problems, we construct and apply operators within explicit and implicit time advance schemes, demonstrating that stability can be robustly guaranteed in each case. We further investigate the use of the linear time advance operators within several integration methods including forward Euler, backward Euler, and the matrix exponential. The application of this method to simulation data from SOLPS-ITER, with varying levels of Markov chain Monte Carlo numerical noise, shows that constrained DMD operators yield a capability to identify, extract, and integrate a (slow) subset of the present timescales. Example applications show that for projected speedup factors of [Formula: see text], and [Formula: see text], a mean relative error of 3%, 5%, and 8% and maximum relative error less than 20% are achievable, which appears acceptable for typical SOLPS-ITER steady-state simulations.
Neural network surrogate models for equations of state
Equation of state (EOS) data provide necessary information for accurate multiphysics modeling, which is necessary for fields such as inertial confinement fusion. Here, we suggest a neural network surrogate model of energy and entropy and use thermodynamic relationships to derive other necessary thermodynamic EOS quantities. We incorporate phase information into the model by training a phase classifier and using phase-specific regression models, which improves the modal prediction accuracy. Our model predicts energy values to 1% relative error and entropy to 3.5% relative error in a log-transformed space. Although sound speed predictions require further improvement, the derived pressure values are accurate within 10% relative error. Our results suggest that neural network models can effectively model EOS for inertial confinement fusion simulation applications.
Latent Twins
Over the past decade, scientific machine learning has transformed the development of mathematical and computational frameworks for analyzing, modeling, and predicting complex systems. From inverse problems to numerical partial differential equations (PDEs), dynamical systems, and model reduction, these advances have pushed the boundaries of what can be simulated. Yet they have often progressed in parallel, with representation learning and algorithmic solution methods evolving largely as separate pipelines. With Latent Twins, we propose a unifying mathematical framework that creates a hidden surrogate in latent space for the underlying equations. Whereas digital twins mirror physical systems in the digital world, Latent Twins mirror mathematical systems in a learned latent space governed by operators. Through this lens, classical modeling, inversion, model reduction, and operator approximation all emerge as special cases of a single principle. We establish the fundamental approximation properties of Latent Twins for both ordinary differential equations (ODEs) and PDEs and demonstrate the framework across three representative settings: (i) canonical ODEs, capturing diverse dynamical regimes; (ii) a PDE benchmark using the shallow-water equations, contrasting Latent Twin simulations with deep operator network and forecasts with a four-dimensional variational method baseline; and (iii) a challenging real-data geopotential reanalysis dataset, reconstructing and forecasting from sparse, noisy observations. Latent Twins provide a compact, interpretable surrogate for solution operators that evaluate across arbitrary time gaps in a single-shot, while remaining compatible with scientific pipelines such as assimilation, control, and uncertainty quantification. Looking forward, this framework offers scalable, theory-grounded surrogates that bridge data-driven representation learning and classical scientific modeling across disciplines.
RE-INTEGRATE EMT Simulation Software: DAE Solvers and Automation
Existing electromagnetic transient (EMT) simulation tools face challenges in accelerating EMT simulations, especially for very large-scale power networks. To tackle this issue, next generation EMT simulation tools such as RE-INTEGRATE EMT are being researched upon. Such tools should be equipped with automation capabilities and advanced numerical differential-algebraic equation (DAE) solvers. In this paper, the DAE solvers incorporated within the RE-INTEGRATE EMT simulation tool are discussed. In particular, a modified ODEINT-based DAE solver and the ARKODE solver from SUN-DIALS are leveraged within RE-INTEGRATE EMT. In addition, the automation implemented within RE-INTEGRATE EMT to automate the DAE generation (replacing the need of manual discretization and assembling DAEs) is discussed. Different use cases were implemented using the RE-INTEGRATE EMT tool and were validated with respect to baseline simulations.
Time-dependent Hamiltonian Simulation of Highly Oscillatory Dynamics and Superconvergence for Schrödinger Equation
We propose a simple quantum algorithm for simulating highly oscillatory quantum dynamics, which does not require complicated quantum control logic for handling time-ordering operators. To our knowledge, this is the first quantum algorithm that is both insensitive to the rapid changes of the time-dependent Hamiltonian and exhibits commutator scaling. Our method can be used for efficient Hamiltonian simulation in the interaction picture. In particular, we demonstrate that for the simulation of the Schrödinger equation, our method exhibits superconvergence and achieves a surprising second order convergence rate, of which the proof rests on a careful application of pseudo-differential calculus. Numerical results verify the effectiveness and the superconvergence property of our method.
Calculation of subdominant solutions of linear differential equations.
Subdominant solutions of linear differential equations calculated by numerical method
Numerical Solution of Symmetric Positive Differential Equations
Finite difference method for symmetric positive linear differential equations, discussing methods to solve finite difference equations generated
Numerical integration of second order differential equations
Performance characteristics of higher order approximations of Runge-Kutta type are analyzed, and performance predictors for time required on machine and for error size are developed. Technique is useful in evaluating system performance, analyzing material characteristics, and designing inertial guidance and nuclear instrumentation and materials.
On the numerical solution of elliptic partial differential equations by the method of lines.
Abstract missing.
Utility of a finite element solution algorithm for initial-value problems
The Galerkin criterion within a finite element Weighted Residuals formulation is employed to establish an implicit solution algorithm for an initial-value partial differential equation. Numerical solutions of a transient parabolic and a hyperbolic equation, obtained using linear, quadratic and two cubic finite element basis functions, are employed to quantize accuracy and confirm and refine theoretical convergence rate estimates. The linear basis algorithm for the hyperbolic equation displays excellent accuracy on a coarse computational grid and a high-order convergence rate with discretization refinement. Good accuracy and a strong convergence rate in surface flux are determined for a nonhomogeneous Neumann boundary constraint applied to a parabolic equation. The results amply demonstrate the impact of the nondiagonal finite element initial-value matrix structure on solution accuracy and/or convergence rate.
Theoretical regime diagrams for thermally driven flows in a beta-plane channel in the presence of variable gravity
The effect of a power law gravity field on baroclinic instability is examined, with a focus on the case of inverse fifth power gravity, since this is the power law produced when terrestrial gravity is simulated in spherical geometry by a dielectric force. Growth rates are obtained of unstable normal modes as a function of parameters of the problem by solving a second order differential equation numerically. It is concluded that over the range of parameter space explored, there is no significant change in the character of theoretical regime diagrams if the vertically averaged gravity is used as parameter.
Higher-Order Panel Method for Aerodynamic Flow Analysis
PANAIR uses high-order panel method to predict inviscid subsonic or supersonic flows about arbitrary configuration. Panel method solves linear partial differential equation numerically by approximating configuration surface with panels on which unknown "singularity strengths" are defined. PANAIR includes advanced software technology as well as advanced aerodynamic technology.
Boundary-layer effects in composite laminates: Free-edge stress singularities, part 6
A rigorous mathematical model was obtained for the boundary-layer free-edge stress singularity in angleplied and crossplied fiber composite laminates. The solution was obtained using a method consisting of complex-variable stress function potentials and eigenfunction expansions. The required order of the boundary-layer stress singularity is determined by solving the transcendental characteristic equation obtained from the homogeneous solution of the partial differential equations. Numerical results obtained show that the boundary-layer stress singularity depends only upon material elastic constants and fiber orientation of the adjacent plies. For angleplied and crossplied laminates the order of the singularity is weak in general.
Analytical calculation of a single jet in cross-flow and comparison with experiment
A series of progressively finer grid systems to calculate the single jet in cross-flow experimentally measured were employed. The experimental measurements provide a collection of velocities, turbulence intensities, and jet concentration profiles with measurements of the inlet field. The use of a series of progressively finer grid systems allows a differentiation between numerical errors and the hydrodynamic modeling assumptions embodied in the 3D combustor code.
A penalty finite element algorithm for parabolic flow problems
The thin-layer simplification of the two-dimensional Navier-Stokes equations for steady viscous flow are developed using an order of magnitude analysis. A space marching finite element solution algorithm is developed, wherein the first order continuity effects are enforced as a penalty function differential constraint. Numerical results are presented to document accuracy and convergence features of the numerical solution algorithm.
An explicit predictor-corrector solver with applications to Burgers' equation
Forward Euler's explicit, finite-difference formula of extrapolation, is used as a predictor and a convex formula as a corrector to integrate differential equations numerically. An application has been made to Burger's equation.
Stability of limit cycles in frictionally damped and aerodynamically unstable rotor stages
This paper deals with the stability of limit cycles (Steady-State Oscillations) associated with the multi-degree-of-freedom model of a frictionally damped and aerodynamically unstable rotor stage. By using the first order averaging technique, a generalized criterion has been established to sort out those unstable limit cycles which govern the maximum transient amplitude beyond which the rotor stage becomes unstable.The stability of the remaining steady-state solutions is analyzed by linearizing the averaged system of differential equations. Numerical results are discussed for three-, four- and five-bladed disks.