Fourth-order Formula Research Articles

Minimum Trotterization Formulas for a Time-Dependent Hamiltonian

When a time propagator eδtA for duration δt consists of two noncommuting parts A=X+Y, Trotterization approximately decomposes the propagator into a product of exponentials of X and Y. Various Trotterization formulas have been utilized in quantum and classical computers, but much less is known for the Trotterization with the time-dependent generator A(t). Here, for A(t) given by the sum of two operators X and Y with time-dependent coefficients A(t)=x(t)X+y(t)Y, we develop a systematic approach to derive high-order Trotterization formulas with minimum possible exponentials. In particular, we obtain fourth-order and sixth-order Trotterization formulas involving seven and fifteen exponentials, respectively, which are no more than those for time-independent generators. We also construct another fourth-order formula consisting of nine exponentials having a smaller error coefficient. Finally, we numerically benchmark the fourth-order formulas in a Hamiltonian simulation for a quantum Ising chain, showing that the 9-exponential formula accompanies smaller errors per local quantum gate than the well-known Suzuki formula.

Improved convergence based on two-point Padé approximants: Simple shear, uniaxial elongation, and flow past a sphere

We investigate the performance of two-point Padé approximants on the truncated series solutions for the shear and elongational viscosities (obtained directly from the exact solutions or by asymptotic methods) for viscoelastic flows modeled with the Giesekus constitutive equation. Τhe major dimensionless groups are the Weissenberg number ( Wi), the Giesekus mobility parameter ( α), and the ratio of the polymer viscosity to the total viscosity of the fluid ( ηp) in terms of which the exact analytical solutions for the shear and elongational viscosities are found. The exact solutions are used to derive truncated series expansions in terms of the Weissenberg number at the Newtonian limit (zero Wi) and in terms of the inverse of the Weissenberg number at the purely elastic limit (infinite Wi). These series are then used to construct low- and high-order two-point diagonal Padé approximants. It is found that both the low- and high-order formulas predict the right trends over the entire range of Wi, from the Newtonian limit to the pure elastic limit. As expected, the high-order formula is much more accurate than the low-order formula. The results show that viscoelastic phenomena, such as shear thinning and extensional thickening, at any value of the Weissenberg number can be predicted analytically with acceptable accuracy using the proposed two-point Padé non-linear analysis. Finally, a fourth-order formula in terms of the Weissenberg number (or in terms of the Deborah number) for the drag force on a rigid sphere that translates with constant speed in a Giesekus fluid is also provided along with a discussion regarding the missing information from the literature, that is, required to perform the proposed non-linear analysis.

PARAMETRIC VARIATION OF AN EXPLICIT FOURTH-STAGE FOURTH-ORDER RUNGE-KUTTA METHOD WITH ABSOLUTE STABILITY

The purpose of this paper is to separate the f(y) functional derivatives by discarding all functional derivatives of f(x,y) after using Taylor Series expansion to expand the fourth-stage fourth-order explicit Runge-Kutta method so as to derive a reduced number of equations for easy computation. Efforts will be made to vary the parameters with the aim of getting a new explicit fourth-order formula that can improve results when implemented on initial-value problems. Efforts will also be made to carry out stability, convergence and consistency analysis and represent the derived equations, their individual f(y) functional derivatives and their various elementary differentials on Butcher’s rooted trees. This idea is derivable from general graphs and combinatorics.

High-Order Deterministic Sensitivity Analysis and Uncertainty Quantification: Review and New Developments

This work reviews the state-of-the-art methodologies for the deterministic sensitivity analysis of nonlinear systems and deterministic quantification of uncertainties induced in model responses by uncertainties in the model parameters. The need for computing high-order sensitivities is underscored by presenting an analytically solvable model of neutron scattering in a hydrogenous medium, for which all of the response’s relative sensitivities have the same absolute value of unity. It is shown that the wider the distribution of model parameters, the higher the order of sensitivities needed to achieve a desired level of accuracy in representing the response and in computing the response’s expectation, variance, skewness and kurtosis. This work also presents new mathematical expressions that extend to the sixth-order of the current state-of-the-art fourth-order formulas for computing fourth-order correlations among computed model response and model parameters. Another novelty presented in this work is the mathematical framework of the 3rd-Order Comprehensive Adjoint Sensitivity Analysis Methodology for Nonlinear Systems (3rd-CASAM-N), which enables the most efficient computation of the exact expressions of the 1st-, 2nd- and 3rd-order functional derivatives (“sensitivities”) of a model’s response to the underlying model parameters, including imprecisely known initial, boundary and/or interface conditions. The 2nd- and 3rd-level adjoint functions are computed using the same forward and adjoint computer solvers as used for solving the original forward and adjoint systems. Comparisons between the CPU times are also presented for an OECD/NEA reactor physics benchmark, highlighting the fact that finite-difference schemes would not only provide approximate values for the respective sensitivities (in contradistinction to the 3rd-CASAM-N, which provides exact expressions for the sensitivities) but would simply be unfeasible for computing sensitivities of an order higher than first-order. Ongoing work will generalize the 3rd-CASAM-N to a higher order while aiming to overcome the curse of dimensionality.

Higher-order formulas of amplitude-dependent tune shift caused by a sextupole magnetic field distribution

Nowadays, designs for ring-based light sources use multibend lattices for achieving a very small emittance of around 100 pmrad. In this type of storage ring, the chromaticity correcting sextupoles generally have greater strengths than those used in typical third-generation light sources. Therefore, controlling lattice nonlinearity such as amplitude-dependent tune shift (ADTS) is important for enabling stable operations and smooth beam commissioning. As the strength of the sextupoles increases, their higher-order terms contribute significantly to ADTS, rendering well-known lowest-order formulas inadequate for describing tune variations at large horizontal amplitudes. In response, we have derived explicit expressions of ADTS up to the fourth order in sextupole strength based on the canonical perturbation theory, assuming that the amplitude of a vertical betatron oscillation is smaller compared with the horizontal one. The new formulas express the horizontal and vertical betatron tune variations as functions of the action variables: ${J}_{x}$ and ${J}_{y}$ up to $O({J}_{x}^{2})$ and $O({J}_{y})$. The derived formulas were applied to a five-bend achromat lattice designed for the SPring-8 upgrade. By comparing the calculated results with the tracking simulations, we found that (1) the formulas accurately express ADTS around a horizontal amplitude of $\ensuremath{\sim}10\text{ }\text{ }\mathrm{mm}$ and (2) the nonlinear terms of the fourth order in sextupole strength govern the behaviors of circulating electrons at large horizontal amplitudes. In this paper, we present explicit expressions of fourth-order formulas of ADTS and provide some examples to illustrate their effectiveness.

High-order finite-volume solutions of the steady-state advection–diffusion equation with nonlinear Robin boundary conditions

We propose high-order finite-volume schemes for numerically solving the steady-state advection–diffusion equation with nonlinear Robin boundary conditions. Although the original motivation comes from a mathematical model of blood clotting, the nonlinear boundary conditions may also apply to other scientific problems. The main contribution of this work is a generic algorithm for generating third-order, fourth-order, and even higher-order explicit ghost-filling formulas to enforce nonlinear Robin boundary conditions in multiple dimensions. Under the framework of finite volume methods, this appears to be the first algorithm of its kind. Numerical experiments on boundary value problems show that the proposed fourth-order formula can be much more accurate and efficient than a simple second-order formula. Furthermore, the proposed ghost-filling formulas may also be useful for solving other partial differential equations.

Evaluation of the fourth-order tesseroid formula and new combination approach to precisely determine gravitational potential

Calculating topographic gravitational potential (GP) is a time-consuming process in terms of efficiency. Prism, mass-point, mass-line, and tesseroid formulas are generally used to calculate the topographic GP effect. In this study, we reformulate the higher-order formula of the tesseroid by Taylor series expansion and then evaluate the fourth-order formula by numerical tests. Different simulation computations show that the fourth-order formula is reliable. Using the conventional approach in numerical calculations, the approximation errors in the areas of the north and south poles are extremely large. Thus, in this study we propose an approach combining the precise numerical formula and tesseroid formulas, which can satisfactorily solve the calculation problem when the computation point is located in the polar areas or areas very near the surface. Furthermore, we suggest a “best matching choice” of new combination approach to calculate the GP precisely by conducting various experiments. Given the computation point at different positions, we may use different strategies. In the low latitude, we use a precise numerical formula, the fourth-order tesseroid formula, the second-order tesseroid formula, and the zero-order formula, in the 1° range (from the computation point), 1° to 15° range, 15° to 40° range, and the range outside 40°, respectively. The accuracy can reach 2 × 10−5 m2 s−2. For the high latitude, we use the precise numerical formula, fourth-order tesseroid, second-order tesseroid, and zero-order tesseroid formulas in the ranges of 0° to 1°, 1° to 10°, 10° to 30°, and the zones outside 30°, respectively. However, if an accuracy level of 2 × 10−5 m2 s−2 is required, the zero-order tesseroid formulas should not be used and the second-order tesseroid formula should be used in the region outside 15° for the low latitude and in the region outside 10° for the high latitude.

On efficient numerical methods for an initial-boundary value problem with nonlocal boundary conditions

Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In this paper, the problem of solving the one-dimensional wave equation subject to given initial and non-local boundary conditions is considered. These non-local conditions arise mainly when the data on the boundary cannot be measured directly. Several finite difference methods with low order have been proposed in other papers for the numerical solution of this one dimensional non-classic boundary value problem. Here, we derive a new family of efficient three-level algorithms with higher order to solve the wave equation and also use a Simpson formula with higher order to approximate the integral conditions. Additionally, the fourth-order formula is also adapted to nonlinear equations, in particular to the well-known nonlinear Klein–Gordon equations which many physical problems can be modeled with. Numerical results are presented and are compared with some existing methods showing the efficiency of the new algorithms.

A Higher-Order Formula of Path Interval for Tool-Path Generation

This paper presents a novel fourth-order formula for determining path intervals and comprehensively considers path interval formulas. In tool-path generation, a path interval is generally formulated as a scallopheight polynomial. Controlling scallop height in mechanical machining improves surface roughness or machining efficiency. We derived a novel fourth-order formula for determining path intervals after reviewing several formulas, then compared formulas. This clarified the differences between path interval formulas with graphic evidence. In micromechanical machining, an approximate expression has an advantage in computational cost but a disadvantage in accuracy. Although our proposed formula includes the fourth order-term scallop height, it requires low computational cost and can be applied to the determining path intervals for free-form surfaces in micromechanical machining. In addition, a correction method of the surface roughness on a free-form surface measured with a profilometer was proposed.

Fourth order correction to a paraxial Gaussian beam in a rotationally symmetric system of non-spherical surfaces

AbstractWe formulate the fourth order correction to a paraxial Gaussian beam propagating through a non-spherical surface system that is rotationally symmetric. First we represent the Gaussian beam by a complex-source-point spherical wave (CSPSW). Next we examine the evolution of the CSPSW through the non-spherical surface system by applying a Fresnel–Kirchhoff diffraction integral. We find a ray-optical solution to the diffraction integral in terms of both the conic constant and the aspheric coefficient of fourth order. Then we numerically evaluate the quality factor of the fourth order-corrected beam propagating through either an aspheric mirror or a thin lens made up of two non-spherical surfaces. The fourth order formulas derived here can be useful in determining the quality factor of the Gaussian beam degraded by a rotationally symmetric system of non-spherical surfaces.

Runge–Kutta pairs of order 5(4) satisfying only the first column simplifying assumption

Among the most popular methods for the solution of the Initial Value Problem are the Runge–Kutta pairs of orders 5 and 4. These methods can be derived solving a system of nonlinear equations for its coefficients. To achieve this, we usually admit various simplifying assumptions. The most common of them are the so-called row simplifying assumptions. Here we neglect them and present an algorithm for the construction of Runge–Kutta pairs of orders 5 and 4 based only in the first column simplifying assumption. The result is a pair that outperforms other known pairs in the bibliography when tested to the standard set of problems of DETEST. A cost free fourth order formula is also derived for handling dense output.

An implicit fourth-order compact finite difference scheme for one-dimensional Burgers’ equation

A fourth-order compact finite difference method is proposed in this paper to solve one-dimensional Burgers’ equation. The newly proposed method is based on the Hopf–Cole transformation, which transforms the original nonlinear Burgers’ equation into a linear heat equation, and transforms the Dirichlet boundary condition into the Robin boundary condition. The linear heat equation is then solved by an implicit fourth-order compact finite difference scheme. A compact fourth-order formula is also developed to approximate the Robin boundary conditions, while the initial condition for the heat equation is approximated using Simpson’s rule to maintain overall fourth-order accuracy. Numerical experiments have been conducted to demonstrate the efficiency and high-order accuracy of this method. The numerical results also show that the method is unconditionally stable, as there is no constraint on time step size.

Path integral ground state study of finite-size systems: Application to small (parahydrogen)N (N=2–20) clusters

We use the path integral ground state method to study the energetic and structural properties of small para-H2 clusters of sizes ranging from 2 to 20 molecules. A fourth order formula is used to approximate the short imaginary-time propagator and two interaction potentials are considered. Our results are compared to those of exact basis set calculations and other quantum Monte Carlo methods when available. We find that for all cluster sizes considered, our results show a lower ground state energy than literature values obtained by diffusion Monte Carlo and variational Monte Carlo. For the dimer and trimer, ground state energies are in good agreement with exact results obtained using the discrete variable representation. Structural properties are found to be insensitive to the choice of interaction potential. We explore the use of Pekeris coordinates to analyze the importance of linear arrangement in trimers and for trimers within clusters of larger size.

High order non-unitary split-step decomposition of unitary operators

We propose a high order numerical decomposition of exponentials of hermitean operators in terms of a product of exponentials of simple terms, following an idea which has been pioneered by M. Suzuki, however implementing it for complex coefficients. We outline a convenient fourth order formula which can be written compactly for arbitrary number of noncommuting terms in the Hamiltonian and which is superiour to the optimal formula with real coefficients, both in complexity and accuracy. We show asymptotic stability of our method for sufficiently small time step and demonstrate its efficiency and accuracy in different numerical models.

Families of accurate discretizations of order two and four for 3-D mildly nonlinear biharmonic problems of second kind

For the numerical integration of the system of three dimensional mildly nonlinear biharmonic problems of second kind, we report two sets of difference methods of order two and four over a cubic domain. The second order derivatives of the solutions are obtained as by-product of the methods. We use 19 and 27 grid points for second and fourth order methods respectively and do not require fictitious points in order to approximate the boundary conditions. We compare the new second and fourth order formulas with the exact solutions both in singular and non-singular cases.

Finite difference methods of order two and four for 2-d non-linear biharmonic problems of first kind

For the numerical integration of the 2-D nonlinear biharmonic problems of first kind, we report two difference methods of order two and four over a rectangular domain. These methods use only the nine grid points and do not require fictitious points in order to approximate the boundary conditions. Derivatives of the solution are obtained as a by-product of the methods. In numerical experiments, the new second and fourth order formulas are compared with the exact solutions.

A method for constructing generalized Runge-Kutta methods

In the implementation of an implicit Runge-Kutta formula, we need to solve systems of nonlinear equations. In this paper, we analyze the Newton iteration process and a modified Newton iteration process for solving these equations. Then we propose the methods in which we take only a fixed finite number of iterations and adopt the last iterate as the approximate solution for the nonlinear equations. Our methods are a kind of generalized Runge-Kutta methods with the same order as the original Runge-Kutta formula and inherit its linear stability properties of the original implicit Runge-Kutta formula, for example, AN-stability, L-stability and S-stability. Based on this fact, we construct three methods of order five imbedding a fourth-order formula for error estimation. Finally, test results for 25 stiff problems are discussed.

Attainable order of theP-stable family of certain two-step methods for periodic second order initial value problems

Certain two-step hybrid methods are considered for periodic second order initial value problems. Some authors have givenP-stable fourth order formula of this type. We shall give a necessary and sufficient condition for the sixth order formula, for whichP-stability analysis is carried out. Attainable order ofP-stable formula of the type is eventually established as four.

A Fifth-Order Exponentially Fitted Formula

A fifth-order second derivative formula is developed for stiff systems of ordinary differential equations. The exponentially fitted formulas possess a large region of absolute stability, and numerical results demonstrate increased accuracy with the same computational effort when compared with similar fourth-order formulas.

Rapid ray tracing in radial gradients

Analytic ray-tracing formulas provide a rapid means of tracing rays, including skew rays, in a radial gradient. A fifth-order formula for the ray position and direction and a fourth-order formula for the optical path length give sufficiently accurate results for various applications.