A method for the numerical integration of coupled first-order differential equations with greatly different time constants.
Numerical integration of coupled first order ODE OF greatly differing time constant
DIFFERENTIAL EQUATION↗
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Magnus method for electronic structure calculations at extreme conditions
We present the application of Magnus based methods to the solution of first order coupled ordinary differential equations in High Energy Density (HED) physics applications. Our focus is on the application to quantum mechanical methods, specifically on the solution of the radial Dirac equation for real and complex energies. HED applications require accurate solutions across a wide range of spatial and energy domains, including regimes where the solutions exhibit pronounced oscillatory behavior. Such cases pose significant computational challenges. We demonstrate that Magnus-based integrators can efficiently and accurately address these challenges. We discuss the implementation of the Magnus method for the solution of the radial Dirac equation, including practical considerations such as the evaluation of matrix exponentials, numerical integration, error estimation, and adaptive step size control. We also discuss the application of these methods to complex energy Green’s function techniques and the efficient approximation of integrals of the solutions relevant to HED electronic structure calculations. Here, we demonstrate the accuracy and robustness of the resulting method in applications to the free-particle case, for which analytic solutions are available for comparison, as well as the challenging case of gold at HED conditions.
general physics↗
Solving the Cauchy-Riemann equations on parallel computers
Discussed is the implementation of a single algorithm on three parallel-vector computers. The algorithm is a relaxation scheme for the solution of the Cauchy-Riemann equations; a set of coupled first order partial differential equations. The computers were chosen so as to encompass a variety of architectures. They are: the MPP, and SIMD machine with 16K bit serial processors; FLEX/32, an MIMD machine with 20 processors; and CRAY/2, an MIMD machine with four vector processors. The machine architectures are briefly described. The implementation of the algorithm is discussed in relation to these architectures and measures of the performance on each machine are given. Simple performance models are used to describe the performance. These models highlight the bottlenecks and limiting factors for this algorithm on these architectures. Conclusions are presented.
Fatoohi, Raad A.↗
Singular perturbation equations for flexible satellites
Force equations of motion of the individual flexible elements of a satellite were obtained in a previous paper. Moment equations of motion of the composite bodies of a flexible satellite are to be developed using two sets of equations which form the basic system for any dynamic model of flexible satellites. This basic system consists of a set of N-coupled, nonlinear, ordinary, or partial differential equations, for a flexible satellite with n generalized, structural position coordinates. For single composite body satellites, N is equal to (n + 3); for dual-spin systems, N is equal to (n + 9). These equations involve time derivatives up to the second order. The study shows a method of avoiding this linearization by reducing the N equations to 3 or 9 nonlinear, coupled, first order, ordinary, differential equations involving only the angular velocities of the composite bodies. The solutions for these angular velocities lead to linear equations in the n generalized structural position coordinates, which can be solved by known methods.
Huang, T. C.↗
Mathematical solutions in internal dose assessment: A comparison of Python-based differential equation solvers in biokinetic modeling
Abstract In biokinetic modeling systems employed for radiation protection, biological retention and excretion have been modeled as a series of discretized compartments representing the organs and tissues of the human body. Fractional retention and excretion in these organ and tissue systems have been mathematically governed by a series of coupled first-order ordinary differential equations (ODEs). The coupled ODE systems comprising the biokinetic models are usually stiff due to the severe difference between rapid and slow transfers between compartments. In this study, the capabilities of solving a complex coupled system of ODEs for biokinetic modeling were evaluated by comparing different Python programming language solvers and solving methods with the motivation of establishing a framework that enables multi-level analysis. The stability of the solvers was analyzed to select the best performers for solving the biokinetic problems. A Python-based linear algebraic method was also explored to examine how the numerical methods deviated from an analytical or semi-analytical method. Results demonstrated that customized implicit methods resulted in an enhanced stable solution for the inhaled 60 Co (Type M) and 131 I (Type F) exposure scenarios for the inhalation pathway of the International Commission on Radiological Protection (ICRP) Publication 130 Human Respiratory Tract Model (HRTM). The customized implementation of the Python-based implicit solvers resulted in approximately consistent solutions with the Python-based matrix exponential method ( expm ). The differences generally observed between the implicit solvers and expm are attributable to numerical precision and the order of numerical approximation of the numerical solvers. This study provides the first analysis of a list of Python ODE solvers and methods by comparing their usage for solving biokinetic models using the ICRP Publication 130 HRTM and provides a framework for the selection of the most appropriate ODE solvers and methods in Python language to implement for modeling the distribution of internal radioactivity.
61 RADIATION PROTECTION AND DOSIMETRY↗
Evolution of coupled weakly driven waves in a dissipative plasma
The nonlinear collisional dynamics of coupled driven plasma waves in the presence of background dissipation is studied analytically within kinetic theory. Sufficiently near marginal stability, phase space correlations are poorly preserved and time delays become unimportant. The system is then shown to be governed by two first-order coupled autonomous differential equations of cubic order for the wave amplitudes and two complementary first-order equations for the evolution of their phases. That system of equations can be decoupled and further simplified to a single second-order differential equation of Liénard's type for each amplitude. Numerical solutions for this equation are obtained in the general case, while analytic solutions are obtained for special cases in terms of parameters related to the spacing of the resonances of the two waves in frequency space, e.g., wave lengths and oscillation frequencies. These parameters are further analyzed to find classes of quasi-steady saturation and pulsating scenarios. To classify equilibrium points, local stability analysis is applied, and bifurcation conditions are determined. When the two waves saturate at similar amplitude levels, their combined signal is shown to invariably exhibit amplitude beating and phase jumps of nearly π. In conclusion, the obtained analytical results can be used to benchmark simulations and to interpret eigenmode amplitude measurements in fusion experiments.
70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗
Nonlinear behavior of acoustic waves in combustion chambers
The nonlinear growth and limiting amplitude of acoustic waves in a combustion chamber are considered. A formal framework is provided within which practical problems can be treated with a minimum of effort and expense. The general conservation equations were expanded in two small parameters, one characterizing the mean flow field and one measuring the amplitude of oscillations, and then combined to yield a nonlinear inhomogeneous wave equation. The unsteady pressure and velocity fields were expressed as syntheses of the normal modes of the chamber, but with unknown time-varying amplitudes. This procedure yielded a representation of a general unsteady field as a system of coupled nonlinear oscillators. The system of nonlinear equations was treated by the method of averaging to produce a set of coupled nonlinear first order differential equations for the amplitudes and phases of the modes. The analysis is applicable to any combustion chamber. The most interesting applications are probably to solid rockets, liquid rockets, or thrust augmentors on jet engines.
Culick, F. E. C.↗
Nonlinear behavior of acoustic waves in combustion chambers. I, II
The general problem of the nonlinear growth and limiting amplitude of acoustic waves in a combustion chamber is treated in three parts: (1) the general conservation equations are expanded in two small parameters, and then combined to yield a nonlinear inhomogeneous wave equation, (2) the unsteady pressure and velocity fields are expressed as a synthesis of the normal modes of the chamber, but with unknown time-varying amplitudes, and (3) the system of nonlinear equations is treated by the method of averaging to produce a set of coupled nonlinear first order differential equations for the amplitudes and phases of the modes. This approximate analysis is applied to the investigation of the unstable motions in a solid propellant rocket engine and in a T burner.
Culick, F. E. C.↗
Comparison of Computed and Measured Performance of a Pulsed Inductive Thruster Operating on Argon Propellant
Pulsed inductive plasma accelerators are electrodeless space propulsion devices where a capacitor is charged to an initial voltage and then discharged through a coil as a high-current pulse that inductively couples energy into the propellant. The field produced by this pulse ionizes the propellant, producing a plasma near the face of the coil. Once a plasma is formed if can be accelerated and expelled at a high exhaust velocity by the Lorentz force arising from the interaction of an induced plasma current and the magnetic field. A recent review of the developmental history of planar-geometry pulsed inductive thrusters, where the coil take the shape of a flat spiral, can be found in Ref. [1]. Two concepts that have employed this geometry are the Pulsed Inductive Thruster (PIT)[2, 3] and the Faraday Accelerator with Radio-frequency Assisted Discharge (FARAD)[4]. There exists a 1-D pulsed inductive acceleration model that employs a set of circuit equations coupled to a one-dimensional momentum equation. The model was originally developed and used by Lovberg and Dailey[2, 3] and has since been nondimensionalized and used by Polzin et al.[5, 6] to define a set of scaling parameters and gain general insight into their effect on thruster performance. The circuit presented in Fig. 1 provides a description of the electrical coupling between the current flowing in the thruster I1 and the plasma current I2. Recently, the model was upgraded to include an equation governing the deposition of energy into various modes present in a pulsed inductive thruster system (acceleration, magnetic flux generation, resistive heating, etc.)[7]. An MHD description of the plasma energy density evolution was tailored to the thruster geometry by assuming only one-dimensional motion and averaging the plasma properties over the spatial dimensions of the current sheet to obtain an equation for the time-evolution of the total energy. The equation set governing the dynamics of the coupled electrodynamic-current sheet system is composed of first-order, coupled ordinary differential equations that can be easily solved numerically without having to resort to much more complex 2-D finite element plasma simulations.
Polzin, Kurt A.↗
Multiple scattering from finite inhomogeneous media
Utilizing the characteristic information concerning the apparent phase constant difference between the electric and magnetic fields propagating in an inhomogeneous medium, a theoretical analysis of the multiple scattering of electromagnetic waves in finite inhomogeneous media is presented. The solution is obtained by first approximating the coefficients of a pair of exact coupled first-order differential equations and then solving the equations by first-order iteration. The present first-order approximate solution with multiple scattering considerations is shown to be more accurate than the WKB solution. Methods to improve the accuracy of the first-order solution further are discussed. Application of the solution to slowly varying finite media with periodic properties demonstrates the validity of the solution. The same approach can be extended to frequencies in the optical region by retaining additional terms in the coefficients of the coupled differential equations.
Tang, C. C. H.↗
On the use of a finite difference method for solving anisotropic scattering problems
A new method of solving the radiative transfer equation is developed in which the scattering and absorption coefficients may have arbitrary variations with depth, and in which both internal (thermal) emission and incident radiation are allowed. Specular and diffuse reflection at both boundaries also is taken into account. The method begins by forming a paired set of coupled first-order differential equations for the symmetric and antisymmetric parts of the radiation field after writing the scattering integral as a numerical quadrature. These differential equations are broken into finite difference form, in which the symmetric and antisymmetric parts of the radiation field are found on alternate grid points. Numerical results for a number of test problems are shown, demonstrating that the method is very fast, that it returns specific intensities and fluxes that are accurate to at least a percent, and that it can be applied to optically thick problems.
Barkstrom, B. R.↗
Analysis of the single-blow transient testing technique for perforated plate heat exchangers
An analysis of the single-blow transient testing technique for heat exchanger surfaces is made for perforated plate and similar discontinuous surfaces. The model assumes that there is no temperature variation across each plate, and allows for axial conduction in spacers: the resulting axial fluid temperature profile is discontinuous, rather than the usual continuous profile. Numerical solutions are obtained to the resulting set of coupled first-order differential equations for a step change in inlet fluid temperature. Results are presented in tabular form, which allow the heat transfer coefficients to be calculated from test data using the maximum slope technique.
Rodriguez, Jose I.↗
Singular perturbation equations for flexible satellites
The dynamic model of a flexible satellite with n generalized structural position coordinates requires the solution of a set of N coupled nonlinear ordinary or partial differential equations. For single composite body satellites, N is equal to (n + 3). For dual-spin systems, N is equal to (n + 9). These equations usually involve time derivatives up to the second order. For large values of n, linearization of the system has so far been the only practicable way of solution. The present study shows a method of avoiding this linearization by reducing the N equations to three or nine nonlinear, coupled, first-order ordinary differential equations involving only the angular velocities of the composite bodies. The solutions for these angular velocities lead to linear equations in the n generalized structural position coordinates, which can then be solved by known methods.
Huang, T. C.↗
A theory of post-stall transients in multistage axial compression systems
A theory is presented for post stall transients in multistage axial compressors. The theory leads to a set of coupled first-order ordinary differential equations capable of describing the growth and possible decay of a rotating-stall cell during a compressor mass-flow transient. These changing flow features are shown to have a significant effect on the instantaneous compressor pumping characteristic during unsteady operation, and henace on the overall system behavior. It is also found from the theory that the ultimate mode of system response, stable rotating stall or surge, depends not only on the B parameter but also on other parameters, such as the compressor length-to-diameter ratio. Small values of this latter quantity tend to favor the occurrence of surge, as do large values of B. A limited parametric study is carried out to show the impact of the different system features on transient behavior. Based on analytical and numerical results, several specific topics are suggested for future research on post-stall transients.
Moore, F. K.↗
A minimum principle for smooth first-order distributed systems
Consideration of the problem of characterizing optimal controls for a class of distributed parameter systems. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables. Boundary conditions on the state are in the form of a finite number of algebraic relations between the state and boundary control variables. A few directions are outlined in which the results obtained may be extended further.
Johnson, T. L.↗
Optimal control of first order distributed systems
The problem of characterizing optimal controls for a class of distributed-parameter systems is considered. The system dynamics are characterized mathematically by a finite number of coupled partial differential equations involving first-order time and space derivatives of the state variables, which are constrained at the boundary by a finite number of algebraic relations. Multiple control inputs, extending over the entire spatial region occupied by the system ("distributed controls') are to be designed so that the response of the system is optimal. A major example involving boundary control of an unstable low-density plasma is developed from physical laws.
Johnson, T. L.↗
Real-time coupled-cluster approach for the cumulant Green's function
Green’s function methods within many-body perturbation theory provide a general framework for treating electronic correlations in excited states. Here we investigate the cumulant form of the one-electron Green’s function based on the coupled-cluster equation of motion approach in an extension of our previous study. The approach yields a non-perturbative expression for the cumulant in terms of the solution to a set of coupled first order, non-linear differential equations. The method thereby adds non-linear corrections to traditional cumulant methods linear in the self energy. The approach is applied to the core-hole Green’s function and illustrated for a number of small molecular systems. For these systems we find that the non-linear contributions lead to significant improvements both for quasiparticle properties such as core-level binding energies, as well as the satellites corresponding to inelastic losses observed in photoemission spectra.
Vila, Fernando D.↗
Equation-of-Motion Coupled-Cluster Cumulant Green’s Function for Excited States and X-ray Spectra
Green’s function methods within many-body perturbation theory provide a general framework for treating electronic correlations in excited states and spectra. Conventional methods using the Dyson equation or the cumulant expansion are typically based on the GW self-energy approximation. In order to extend this approximation in molecular systems, a non-perturbative real-time coupled-cluster cumulant Green’s function approach has been introduced, where the cumulant is obtained as the solution to a set of coupled first order, non-linear differential equations. This approach naturally includes non-linear corrections to conventional cumulant Green’s function techniques where the cumulant is linear in the GW self-energy. The method yields the spectral function associated with the core Green’s function, which is directly related to the x-ray photoemission spectra (XPS) of molecular systems. The approach also yields very good results for binding energies and satellite excitations. The x-ray absorption spectrum (XAS) is then obtained as a convolution of the core spectral function and an effective one-body absorption spectrum. Here this approach is extended to include the full coupled-cluster-singles (CCS) core Green’s function by including the complete form of the non-linear contributions to the cumulant as well as all single, double, and triple cluster excitations in the CC amplitude equations. This approach naturally builds in orthogonality and shake-up effects analogous to those in the Mahan-Noizeres-de Dominicis edge singularity corrections that enhance the XAS near the edge. Themethod is illustrated for the XPS and XAS of NH 3 .
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗