Algebraic Order Research Articles

Runge–Kutta Embedded Methods of Orders 8(7) for Use in Quadruple Precision Computations

High algebraic order Runge–Kutta embedded methods are commonly used when stringent tolerances are demanded. Traditionally, various criteria are satisfied while constructing these methods for application in double precision arithmetic. Firstly we try to keep the magnitude of the coefficients low, otherwise we may experience loss of accuracy; however, when working in quadruple precision we may admit larger coefficients. Then we are able to construct embedded methods of orders eight and seven (i.e., pairs of methods) with even smaller truncation errors. A new derived pair, as expected, is performing better than state-of-the-art pairs in a set of relevant problems.

Runge–Kutta–Nyström methods of eighth order for addressing Linear Inhomogeneous problems

Second order Linear Inhomogeneous Initial Value Problems with constant coefficients are considered here. Runge–Kutta–Nyström (RKN) methods are amongst the most effective for addressing these problems. For attaining a certain algebraic order of accuracy, we need to solve a reduced set of order conditions (in comparison with conventional RKN) with respect to the coefficients. Here, using a seven stages method we manage to solve these equations up to eighth order. Differential Evolution technique was used for achieving this. Thus, our main contribution is that we saved one stage since conventional eight order RKN methods use eight stages for advancing a step. This new method is combined at no cost with a fifth and third order companion formulas. As consequence, a triplet of orders 8(5)3 is given for estimating local errors and adjust effectively the step lengths. The new triplet of methods is applied to various relevant problems and is found to outperform standard RKN pairs from the literature.

Numerical simulation of discretized second-order variable coefficient elliptic PDEs by a Classical Eight-step Model

This article presents a new numerical model with nodes within the interval of eight-step for numerical simulation of a class of variable coefficient elliptic partial differential equations in the two-dimensional domain. The method is developed via the principle of collocation and interpolation techniques using Hermite polynomial as the basis function. The main classical model and its derivatives generated from the continuous function are then united together to form the required Classical Eight-step Model (CEM). The analysis of the CEM was investigated and shows that it satisfies the conditions for convergence with an algebraic order of nine. The CEM is applied to solve the semi-discretized elliptic partial differential equations with a variable coefficient arising from the discretization of one of their spatial variables. The performance of the CEM was established with six test problems. The approximate solution generated using CEM is compared with the analytical solution of the problems and other existing methods in the literature. Numerical illustrations demonstrate that the method is convergent and highly accurate. The advantages of CEM over other existing methods reveal the accuracy and efficiency of the CEM as depicted in curves.

A trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström pair to solve oscillating systems

In this study, a trigonometrically adapted 6(4) explicit Runge–Kutta–Nyström (RKN) pair with six stages is formulated, considering a previous method developed by El‐Mikkawy and Rahmo. The obtained adapted pair integrates exactly the usual test equation: . The local truncation error of the new method is presented, showing that the algebraic order of the original method is maintained. The periodicity interval of the new method is computed, showing that the developed method is “almost” P‐stable. The numerical examples considered clearly show the superiority of the new developed embedded pair over other RKN methods of algebraic orders 6(4) with six stages appeared in the literature.

Numerical Solution to Sixth Order Ordinary Differential Equation Using Three Stage Eighth Order Runge-Kutta Type Method

The present paper provides an innovative approach in developing numerical structure with minimal processing and maximum precision for sixth order ordinary differential equation satisfying underlying associated initial conditions. The algebraic order conditions for the three stage eighth order Runge-Kutta methods (RKSD8) been well designed for evaluating local and global truncated errors towards sixth order ordinary differential equation having form vvi = f (u,v,v',v''). The convergence of a numerical method is well explained and proved beneficial in evaluating norms of error and zero stability of RKSD8 with effective values of weights and nodes in the form of Butcher tableau.

Energy-preserving methods for guiding center system based on averaged vector field

We propose a family of energy-preserving methods for guiding center dynamics by perceiving its Hamiltonian nature based on the averaged vector field. The energy conservation, symmetric property, and algebraic order of these methods are studied. Furthermore, higher order energy-preserving methods are systematically introduced by using a composition technique. Two second order and two fourth order symmetric energy-preserving methods are constructed and applied to simulate the guiding center motion in both the dipole magnetic field and the tokamak magnetic field. Numerical results show that these methods have significant superiorities in energy conservation compared with the existing canonicalized symplectic methods of the corresponding orders. The numerical case of the guiding center motion in the toroidal acceleration electric field exhibits favorable long-term conservative properties of the new methods to the particle-field system, while the kinetic energy of guiding centers keeps increasing. These energy-preserving methods based on the averaged vector field can be applied to any non-canonical Hamiltonian system.

Asymmetric phase diagrams, algebraically ordered Berezinskii-Kosterlitz-Thouless phase, and peninsular Potts flow structure in long-range spin glasses.

The Ising spin-glass model on the three-dimensional (d=3) hierarchical lattice with long-range ferromagnetic or spin-glass interactions is studied by the exact renormalization-group solution of the hierarchical lattice. The chaotic characteristics of the spin-glass phases are extracted in the form of our calculated, in this case continuously varying, Lyapunov exponents. Ferromagnetic long-range interactions break the usual symmetry of the spin-glass phase diagram. This phase-diagram symmetry breaking is dramatic, as it is underpinned by renormalization-group peninsular flows of the Potts multicritical type. A Berezinskii-Kosterlitz-Thouless (BKT) phase with algebraic order and a BKT-spin-glass phase transition with continuously varying critical exponents are seen. Similarly, for spin-glass long-range interactions, the Potts mechanism is also seen, by the mutual annihilation of stable and unstable fixed distributions causing the abrupt change of the phase diagram. On one side of this abrupt change, two distinct spin-glass phases, with finite (chaotic) and infinite (chaotic) coupling asymptotic behaviors are seen with a spin-glass to spin-glass phase transition.

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

Algorithms of algebraic order nine for numerically solving second-order boundary and initial value problems in ordinary differential equations

Sixth Order Numerov-Type Methods with Coefficients Trained to Perform Best on Problems with Oscillating Solutions

Numerov-type methods using four stages per step and sharing sixth algebraic order are considered. The coefficients of such methods are depended on two free parameters. For addressing problems with oscillatory solutions, we traditionally try to satisfy some specific properties such as reduce the phase-lag error, extend the interval of periodicity or even nullify the amplification. All of these latter properties come from a test problem that poses as a solution to an ideal trigonometric orbit. Here, we propose the training of the coefficients of the selected family of methods in a wide set of relevant problems. After performing this training using the differential evolution technique, we arrive at a certain method that outperforms the other ones from this family in an even wider set of oscillatory problems.

Constructing Steklov-type cubature formulas for a finite element in the shape of a bipyramid

This paper reports the construction of cubature formulas for a finite element in the form of a bipyramid, which have a second algebraic order of accuracy. The proposed formulas explicitly take into consideration the parameter of bipyramid deformation, which is important when using irregular grids. The cubature formulas were constructed by applying two schemes for the location of interpolation nodes along the polyhedron axes: symmetrical and asymmetrical. The intervals of change in the elongation (compression) parameter of a bipyramid semi-axis have been determined, within which interpolation nodes of the constructed formulas belong to the integration region, while the weight coefficients are positive, which warrants the stability of calculations based on these cubature formulas. If the deformation parameter of the bipyramid is equal to unity, then both cubature formulas hold for the octahedron and have a third algebraic order of accuracy. The resulting formulas make it possible to find elements of the local stiffness matrix on a finite element in the form of a bipyramid. When calculating with a finite number of digits, a rounding error occurs, which has the same order for each of the two cubature formulas. The intervals of change in the elongation (compression) parameter of the bipyramid semi-axis have been determined, which meet the requirements, which are employed in the ANSYS software package, for deviations in the volume of the bipyramid from the volume of the octahedron. Among the constructed cubature formulas for a bipyramid, the optimal formula in terms of the accuracy of calculations has been chosen, derived from applying a symmetrical scheme of the arrangement of nodes relative to the center of the bipyramid. This formula is invariant in relation to any affinity transformations of the local bipyramid coordinate system. The constructed cubature formulas could be included in libraries of methods for approximate integration used by those software suites that implement the finite element method.

A new implicit symmetric method of sixth algebraic order with vanished phase-lag and its first derivative for solving Schrödinger's equation

Abstract In this article, we have developed an implicit symmetric four-step method of sixth algebraic order with vanished phase-lag and its first derivative. The error and stability analysis of this method are investigated, and its efficiency is tested by solving efficiently the one-dimensional time-independent Schrödinger’s equation. The method performance is compared with other methods in the literature. It is found that for this problem the new method performs better than the compared methods.

A new economical method with eliminated phase-lag and its derivative for problems in chemistry

A new EcoFinDif (Economical Finite Difference) method with eliminated phase-lag and its first derivative, for periodical and/or oscillating initial or boundary value problems with an application on problems in Chemistry is presented in this paper. The method is considered economical since it is of fourteenth algebraic order and uses the minimum number of function evaluations per step.

A Family of Functionally-Fitted Third Derivative Block Falkner Methods for Solving Second-Order Initial-Value Problems with Oscillating Solutions

One of the well-known schemes for the direct numerical integration of second-order initial-value problems is due to Falkner. This paper focuses on the construction of a family of adapted block Falkner methods which are frequency dependent for the direct numerical solution of second-order initial value problems with oscillatory solutions. The techniques of collocation and interpolation are adopted here to derive the new methods. The study of the properties of the proposed adapted block Falkner methods reveals that they are consistent and zero-stable, and thus, convergent. Furthermore, the stability analysis and the algebraic order conditions of the proposed methods are established. As may be seen from the numerical results, the resulting family is efficient and competitive compared to some recent methods in the literature.

Formulation of a New Implicit Method for Group Implicit BBDF in Solving Related Stiff Ordinary Differential Equations

This paper proposed a new alternative approach of the implicit diagonal block backward differentiation formula (BBDF) to solve linear and nonlinear first-order stiff ordinary differential equations (ODEs). We generate the solver by manipulating the numbers of back values to achieve a higher-order possible using the interpolation procedure. The algorithm is developed and implemented in C ++ medium. The numerical integrator approximates few solution points concurrently with off-step points in a block scheme over a non-overlapping solution interval at a single iteration. The lower triangular matrix form of the implicit diagonal causes fewer differentiation coefficients and ultimately reduces the execution time during running codes. We choose two intermediate points as off-step points appropriately, which are proven to guarantee the method's zero stability. The off-step points help to increase the accuracy by optimizing the local truncation error. The proposed solver satisfied theoretical consistency and zero-stable requirements, leading to a convergent multistep method with third algebraic order. We used the well-known and standard linear and nonlinear stiff IVP problems used in literature for validation to measure the algorithm's accuracy and processor time efficiency. The performance metrics are validated by comparing them with a proven solver, and the output shows that the alternative method is better than the existing one.

Numerical simulation of second-order initial-value problems using a new class of variable coefficients and two-step semi-hybrid methods

In this paper, a new family of two-step semi-hybrid schemes of the 12th algebraic order is proposed for the numerical simulation of initial-value problems of second-order ordinary differential equations. The proposed methods are symmetric and belong to the family of multiderivative methods. Each method of the new family appears to be hybrid, but after implementing the hybrid terms, it will continue as a multiderivative method. Therefore, the designation semi-hybrid is used. The consistency, convergence, stability, and periodicity of the methods are investigated and analyzed. In order to show the accuracy, consistency, convergence, and stability of the proposed family, it was tested on some well-known problems, such as the undamped Duffing’s equation. The simulation results demonstrate the efficiency and advantages of the proposed method compared to the currently available methods.

Symplectic Runge-Kutta-Nystr ¨ om Methods with Phase-Lag Oder 8 and Infinity

In this work we consider Symplectic Runge Kutta Nystrom methods with five stages. A new fourth algebraic order method with phase-lag order eight is presented. Also the symplectic Runge Kutta Nystrom of Calvo and Sanz Serna with five stages and fourth order is modified to produce a phase-fitted method. We apply th e new methods on several Hamiltonian systems and on the computation of the eigenvalues of the Schrodinger Equation.

Spatiotemporal scaling of two-dimensional nonequilibrium exciton-polariton systems with weak interactions

We perform a theoretical study on the spatiotemporal correlation of two-dimensional (2D) nonequilibrium exciton-polariton (EP) systems driven by incoherent pumping. Based on the stochastic generalized Gross-Pitaevskii equation, we calculate the density fluctuation, coherence function, and scaling function. It is found that the correlations of short range agree well with the Bogoliubov linear theory, justifying the existence of an algebraic order. At large distances, both dynamic and static correlations are characterized by the nonlinear scaling behaviors of the Kardar-Parisi-Zhang universality class, especially when the interaction is weak. The coexistence of different types of correlations with their respective length scales is a distinctive feature of 2D EP systems. In addition, the interplay between vortices is modified in the regime of weak interaction and strong pumping, leading to complex nonequilibrium vortex patterns.

A Trigonometrically Adapted Embedded Pair of Explicit Runge-Kutta-Nyström Methods to Solve Periodic Systems

In this paper a 5(3) pair of explicit trigonometrically adapted Runge-Kutta-Nystrom methods with four stages is derived based on an explicit pair appeared in the literature. The new adapted method is able to integrate exactly the usual test equation: y′′ = −w 2 y. The local truncation error of the new method is obtained, proving that the algebraic order of convergence is maintained. The stability interval of the new method is obtained, showing that the proposed method is absolutely stable. The numerical experiments performed demonstrate the robustness of the new embedded pair in comparison with some standard codes available in the literature.

Spectral Mesh-Free Quadrature for Planar Regions Bounded by Rational Parametric Curves

This work presents spectral, mesh-free, Green’s theorem-based numerical quadrature schemes for integrating functions over planar regions bounded by rational parametric curves. Our algorithm proceeds in two steps: (1) We first find intermediate quadrature rules for line integrals along the region’s boundary curves corresponding to Green’s theorem. (2) We then use a high-order quadrature rule to compute the numerical antiderivative of the integrand along a coordinate axis, which is used to evaluate the Green’s theorem line integral. We present two methods to compute the intermediate quadrature rule. The first is spectrally accurate (it converges faster than any algebraic order with respect to number of quadrature points) and is relatively easy to implement, but has no guarantee of polynomial exactness. The second guarantees exactness for polynomial integrands up to a pre-specified degree k with an a priori-known number of quadrature points and retains the convergence properties of the first, but is slightly more complicated. The quadrature schemes have applications to computation of geometric moments, immersogeometric analysis, conservative field transfer between high-order meshes, and initialization of multi-material simulations with rational geometry. We compare the quadrature schemes produced using our method to other methods in the literature and show that they are much more efficient both in terms of number of quadrature points and computational time. We provide an open-source implementation of the algorithm in MATLAB.

Modulating functions based differentiator of the pseudo-state for a class of fractional order linear systems

In this paper, an algebraic and robust fractional order differentiator is designed for a class of fractional order linear systems with an arbitrary differentiation order in [0,2]. It is designed to estimate the fractional derivative of the pseudo-state with an arbitrary differentiation order as well as the one of the output. In particular, it can also estimate the pseudo-state. Different from our previous works, the considered system no longer relies on the matching conditions, which makes the system model be more general. First, the considered system is transformed into a fractional differential equation from the pseudo-state space representation. Second, based on the obtained equation, a series of equations are constructed by applying different fractional derivative operators. Then, the fractional order modulating functions method is introduced to recursively give algebraic integral formulas for a set of fractional derivatives of the output and a set of fractional derivative initial values. These formulas are used to non-asymptotically and robustly estimate the fractional derivatives of the pseudo-state and the output in discrete noisy cases. Third, the required modulating functions are designed. After giving the associated estimation algorithm, numerical simulation results are finally given to illustrate the accuracy and robustness of the proposed method.