Frequency evaluation for exponentially fitted Runge–Kutta methods

In this study, a comparative analysis of two Runge-Kutta methods; fourth-order Runge-Kutta method and Butcher’s Fifth Order Runge-Kutta method are presented and used to solve systems of first-order linear Ordinary Differential Equations (ODEs). The main interest of this work is to test the accuracy, convergence rate and computational efficiency of these methods by using different numerical problems of ODEs. Empirical conclusions are drawn after close observation of the results presented by the two methods, which further highlights their limitations and enabling researchers to make informed decisions in choosing the appropriate technique for specific systems of ODEs problems.

A first-order ordinary differential equation (ODE) is a function with two variables defined in the xy-axis of a field. Various numerical methods, such as the Euler method, Runge-Kutta method, Heun’s method and others, are used to solve ODEs, with varying computational costs and accuracy. The Euler method can only solve the first derivative equation with the simplest implementation at the lowest cost of computation, but it produces less accurate results. This research focuses on improving the Euler method to increase its accuracy. A new scheme called Centroidal-Polygon (CP) is used in this study. The CP scheme is tested on the Resistor-Capacitor (RC) circuit equation to ensure that it can be used in fields other than mathematics and computation. The RC circuit equation is used to compute maximum error and assess the accuracy of the CP scheme and its counterparts. The circuit equation’s accuracy in the RC circuit equation is determined by the time constant (τ). This research used Scilab 6.0 software to analyze the maximum error. The performance of the CP scheme was compared to the Polygon, Harmonic-Polygon, and Cube-Polygon schemes, which are all enhanced Euler methods. The results show that the CP scheme achieves higher accuracy while requiring less computing time. In future studies, the CP scheme will be applied to the RCL circuit equation and second-order ODE to ensure the CP scheme can be used in all applications.

This article extends the theory of classical finite-difference\nsummation-by-parts (FD-SBP) time-marching methods to the generalized\nsummation-by-parts (GSBP) framework. Dual-consistent GSBP time-marching methods\nare shown to retain: A and L-stability, as well as superconvergence of integral\nfunctionals when integrated with the quadrature associated with the\ndiscretization. This also implies that the solution approximated at the end of\neach time step is superconvergent. In addition GSBP time-marching methods\nconstructed with a diagonal norm are BN-stable. This article also formalizes\nthe connection between FD-SBP/GSBP time-marching methods and implicit\nRunge-Kutta methods. Through this connection, the minimum accuracy of the\nsolution approximated at the end of a time step is extended for nonlinear\nproblems. It is also exploited to derive conditions under which nonlinearly\nstable GSBP time-marching methods can be constructed. The GSBP approach to time\nmarching can simplify the construction of high-order fully-implicit Runge-Kutta\nmethods with a particular set of properties favourable for stiff initial value\nproblems, such as L-stability. It can facilitate the analysis of fully discrete\napproximations to PDEs and is amenable to to multi-dimensional spcae-time\ndiscretizations, in which case the explicit connection to Runge-Kutta methods\nis often lost. A few examples of known and novel Runge-Kutta methods associated\nwith GSBP operators are presented. The novel methods, all of which are L-stable\nand BN-stable, include a four-stage seventh-order fully-implicit method, a\nthree-stage third-order diagonally-implicit method, and a fourth-order\nfour-stage diagonally-implicit method. The relative efficiency of the schemes\nis investigated and compared with a few popular non-GSBP Runge-Kutta methods.\n

In this paper, a new class of two-derivative two-step Runge-Kutta (TDTSRK) methods for the numerical solution of non-stiff initial value problems (IVPs) in ordinary differential equation (ODEs) is considered. The TDTSRK methods are a special case of multi-derivative Runge-Kutta methods proposed by Kastlunger and Wanner (1972). The methods considered herein incorporate only the first and second derivatives terms of ODEs. These methods possess large interval of stability when compared with other existing methods in the literature. The experiments have been performed on standard problems, and comparisons were made with some standard explicit Runge-Kutta methods in the literature.

B-theory of general linear methods for Volterra functional differential equations

We consider the equivalence problem for first-order ordinary differential equations (ODEs) and recover the well-known result that any two first-order ODEs are equivalent under a point transformation by Cartan’s equivalence method. Curvature properties of some first-order and second-order ODEs motivate us to define the equivalence problem for such equations as the metric equivalence problem which defines the [Formula: see text]-structure on the corresponding manifolds. Accordingly, we show that there are first-order ODEs and linear second-order ODEs which may not be equivalent under a class of fiber-preserving diffeomorphisms.

Diagonally implicit Runge–Kutta methods for stiff ODEs

The graded-index (GRIN) media have been widely used in modern optical systems, such as imaging lenses, optical beam delivery components, and beam shaping elements. Accurate and efficient modeling of electromagnetic fields propagating in GRIN media is essential for developing innovative and high-quality optical products. It is the goal of this thesis. We first give an overview of several existing modeling techniques, including the rigorous ones to model symmetric GRIN structures, e.g., Mie theory (spherically symmetric) and several beam propagation methods (BPMs) with approxiamations to model general GRIN structures, which are numerically more efficient than rigorous methods. However, many of BPMs are limited by considering only a small variance of the refractive index. To overcome the limitations, we develop a unified field solver, known as the Runge-Kutta (RK) k-domain field propagation method. It can be used to calculate general input fields propagating through an arbitrary GRIN medium, accurately, without physical approximations, such as the scalar field approximation or the paraxial approximation. We convert Maxwells equations in spatial (x-) domain into ordinary differential equations (ODEs) in the spatial-frequency (k-) domain, which can be solved iteratively using the RK method. In our numerical calculation, taking advantage of the convolution theorem, the total numerical effort is linear in the number of sampling points. As many GRIN components are modeled or designed using ray tracing techniques, the field passing through the GRIN components must show negligible diffraction, which satisfies the geometric field assumption. By substituting geometric field ansatz into the ODEs of RK k-domain field propagation method and Maxwells equations, we obtain two ODEs in x-domain. After solving them by using again RK method, the RK x-domain field propagation method is established.

First-order ordinary differential equations have some rather special properties, which result for the most part because they can only contain a limited number of terms. In fact, all linear first-order ordinary differential equations can easily be solved. This is in contrast to higher-order ordinary differential equations that become much more difficult to solve when, for example, they contain variable coefficients. Furthermore, some methods exist to solve a pretty large class of nonlinear first-order ordinary differential equations, which is generally not the case for higherorder nonlinear equations.

This research paper deals with a numerical algorithm based on the RCW method to solve a set of first-order ordinary differential equations. In this algorithm, the answer of each equation is considered as a polynomial of degree 2. The coefficients used in this polynomial are categorized into groups of free and fixed coefficients. The free coefficients are obtained by optimizing the error function, while the fixed coefficients are computed from the magnitudes of derivative of this polynomial at initial value. To check the correctness of the current algorithm, its results for several case problems are compared with the findings of the Runge-Kutta (RK) method. This comparison shows that the algorithm presented in this paper has a higher accuracy and stability than the RK method even for the nonlinear problems.

In this work, stabilized explicit integrators for local parametrization are introduced to calculate the dynamics of constrained multi-rigid-body systems, including those based on the orthogonal Runge–Kutta–Chebyshev (RKC) method and the extrapolated stabilized explicit Runge–Kutta (ESERK) method. Both of these methods have large stability regions at the negative real axis, and this property makes them suitable to settle the introduction of the stabilization parameter for a constraint equation. The local vectorial rotation parameters are adopted to describe rotations in each rigid body, and a stabilization technique is developed to transform the differential-algebraic equations (DAE) into a set of first-order ordinary differential equations (ODEs) that can be computed efficiently. Several benchmarks are calculated and the results are compared to those by the generalized-α integrator and ADAMS models, verifying their effectiveness in nonstiff problems.

The main aim of this paper was to find the correct method of calculating equations of heat and mass transfer for the adsorption process and to calculate it numerically in reasonable time and with proper accuracy. An adsorption heat pump with a silica gel adsorbent and water adsorbate is discussed. We developed a mathematical model of temperature and uptake changes in the adsorber/desorber comprising the set of heat and mass balance partial differential equations (PDEs), together with the initial and boundary conditions and solved it by the numerical method of lines (NMOL). Spatial discretization was performed with equally spaced axial nodes and the PDEs were reduced to a set of ordinary differential equations (ODEs). We focused on the comparison of results obtained when the set of heat and mass balance ODEs for an adsorber was solved using: (1) the Runge–Kutta fixed step size fourth-order method (RKfixed), (2) the Runge–Kutta–Fehlberg 4.5th-order method with a variable step size (RK45), and (3) the Gear Backward Differentiation Formulae numerical (Gear BDF) methods. In our experience, all three types of ODE numerical methods (RKfixed, RK45, and Gear BDF) can be applied in simple models to model an adsorber with attention on their limitations. The Gear BDF method usually requires much fewer steps than the RK45 method for almost the same calculating time. RK methods require many more steps to obtain results, and the calculating time depends on accuracy or defined time step. Moreover, one should pay attention to the number of nodes or possible oscillations.

This article looks into the use of the fourth-order Runge-Kutta (RK4) method in realistic motion simulation within computer graphics. With dynamic animations, there is an emerging need to solve physical systems using ordinary differential equations, for which RK4 is particularly useful due to its accuracy, stability, and balanced computational cost and efficiency. We implement motion phenomena with damped spring-mass systems by changing second-order differential equations into first-order systems that can be integrated using RK4. The results are measured against Euler and Midpoint methods for assessing stability, error control, and visual smoothness. In every instance, RK4 was found to be the most accurate, stable, and free from overshoot and jitter artifacts. The method demonstrates its effectiveness in real-time animation simulation, including but not limited to simulating cloth movement, flexible body motion, and character dynamic movements through progressive simulation and case studies. Even after undergoing extensive simulation durations, RK4 repeated proved to be supremely reliable regarding energy conservation, damping precision, and fidelity. While perhaps more costly in terms of computation than the more straightforward methods, RK4 remains highly tenable with today's processing capabilities. In addition to greatly improving the physics realism in simulations, the method is commendably applicable in Unity and PhysX or Bullet visual engines. The study illustrates smooth and realistic animation with the help of RK4 while further establishing its importance as a fundamental method for motion graphics and simulation in academic research and industry use.

Hybrid methods for radiation transport using diagonally implicit Runge–Kutta and space–time discontinuous Galerkin time integration

One can elucidate integrability properties of ordinary differential equations (ODEs) by knowing the existence of second integrals (also known as weak integrals or Darboux polynomials for polynomial ODEs). However, little is known about how they are preserved, if at all, under numerical methods. Here, we leverage the recently discovered theory of discrete second integrals to show novel results about Runge-Kutta methods. In particular, we show that any Runge-Kutta method preserves all affine second integrals but cannot preserve all quadratic second integrals of an ODE. A number of interesting corollaries are also discussed, such as the preservation of certain rational integrals by Runge-Kutta methods. The special case of affine second integrals with constant cofactor are also discussed as well the preservation of third and higher integrals.