Global Truncation Error Research Articles

SOLUTION OF ORDINARY DIFFERENTIAL EQUATION vvi (u)=f(u,v,v',v'',v''') USING EIGHTH AND NINTH ORDER RUNGE-KUTTA TYPE METHOD

The present paper presents the numerical conclusion to solve sixth order initial value ordinary differential equation (ODE). The concept of order conditions for three stage eighth order (RKSD8) & four stage ninth order Runge-Kutta methods (RKSD9) has been derived for finding global truncation error of differential equation The global and local truncated errors norms, zero stability of extended Runge-Kutta method (RK) is well defined and demonstrated with the help of an example.

Some New Time and Cost Efficient Quadrature Formulas to Compute Integrals Using Derivatives with Error Analysis

In this research, some new and efficient quadrature rules are proposed involving the combination of function and its first derivative evaluations at equally spaced data points with the main focus on their computational efficiency in terms of cost and time usage. The methods are theoretically derived, and theorems on the order of accuracy, degree of precision and error terms are proved. The proposed methods are semi-open-type rules with derivatives. The order of accuracy and degree of precision of the proposed methods are higher than the classical rules for which a systematic and symmetrical ascendancy has been proved. Various numerical tests are performed to compare the performance of the proposed methods with the existing methods in terms of accuracy, precision, leading local and global truncation errors, numerical convergence rates and computational cost with average CPU usage. In addition to the classical semi-open rules, the proposed methods have also been compared with some Gauss–Legendre methods for performance evaluation on various integrals involving some oscillatory, periodic and integrals with derivative singularities. The analysis of the results proves that the devised techniques are more efficient than the classical semi-open Newton–Cotes rules from theoretical and numerical perspectives because of promisingly reduced functional cost and lesser execution times. The proposed methods compete well with the spectral Gauss–Legendre rules, and in some cases outperform. Symmetric error distributions have been observed in regular cases of integrands, whereas asymmetrical behavior is evidenced in oscillatory and highly nonlinear cases.

L-stable and A-stable numerical method of order two for stiff differential equation

This article presents a finite difference method of order two for stiff differential equation that will overcome the effects of stiffness since it is A-stable. And, reflect the asymptotic behavior of the solution of the stiff problem since it is L-stable. The classical explicit Runge–Kutta methods of order two are not suitable for stiff problems since the numerical solution will not overcome the effects of stiffness and will not reflect the asymptotic behavior of the solution of the stiff problem (for example, Dahlquist problem). And, they are not both A-stable and L-stable. On applying Euler’s implicit method, the numerical solution will overcome the effects of stiffness and will reflect the asymptotic behavior of the solution of the stiff problem. And hence, it is both A-stable and L-stable. But it is of order one method. A new numerical method which is both A-stable and L-stable and of order two for the numerical solution of a stiff differential equation is presented in this article. It will overcome the effects of stiffness and reflects the asymptotic behavior of the solution. The new method works well for large values of step size and works well in large domain. The new method is a modified form of the mid-point rule for the stiff problems. The rate of order of convergence is proved to be two both theoretically and numerically. Experimental results show the performance of the method based on the metrics such as stability function, stability region, order star fingers, the rate of order of theoretical and numerical convergence, absolute and relative errors, percentage of numerical solution accuracy and local and global truncation errors both numerically and graphically.

Solving Fractional Fuzzy Impulsive Differential Equations with Uncertainty by Novel Computational Technique

The aim of this paper is to utilize the fuzzy fractional generalized Taylor series for fuzzy fractional impulsive differential equations (FFIDE) with uncertainty in the sense for generalized Hukuhara differentiability. Then, for the FFIDE, the modified fuzzy fractional Euler technique (MFFET) is presented following the fuzzy fractional generalized Taylor series and its local and global truncation errors are defined. Furthermore, the consistency, convergence, and stability for this MFFET are provided in detail. The illustrative examples show that the above technique, owing to its usefulness and efficiency, is used for solving [Formula: see text]th-order nonlinear FFIDES.

Selecting the load at the intermediate time point of the ρ∞-Bathe time integration scheme

The objective of this paper is to identify the optimal load selection at the intermediate time point of the ρ∞-Bathe time integration method. We study the truncation errors of the scheme in homogeneous and forced responses for various parameters. The optimal load at the sub-step is determined by minimizing the global truncation errors of forced responses. A numerical impulse analysis shows that the optimal load at the sub-step thus established actually corresponds to numerical impulses at the three- and four-point Newton-Cotes formulas for the second- and third-order accuracy cases, respectively. We illustrate the findings of our theoretical study in example solutions of two-dimensional structural dynamics and wave propagation problems. With the optimally selected load at the sub-step, more accurate solutions can be obtained in some analyses.

A Fuzzy Method for Solving Fuzzy Fractional Differential Equations Based on the Generalized Fuzzy Taylor Expansion

In this field of research, in order to solve fuzzy fractional differential equations, they are normally transformed to their corresponding crisp problems. This transformation is called the embedding method. The aim of this paper is to present a new direct method to solve the fuzzy fractional differential equations using fuzzy calculations and without this transformation. In this work, the fuzzy generalized Taylor expansion by using the sense of fuzzy Caputo fractional derivative for fuzzy-valued functions is presented. For solving fuzzy fractional differential equations, the fuzzy generalized Euler’s method is introduced and applied. In order to show the accuracy and efficiency of the presented method, the local and global truncation errors are determined. Moreover, the consistency, convergence, and stability of the generalized Euler’s method are proved in detail. Eventually, the numerical examples, especially in the switching point case, show the flexibility and the capability of the presented method.

Fast and High-Order Accuracy Numerical Methods for Time-Dependent Nonlocal Problems in $${\pmb {\mathbb {R}}}^2$$

In this paper, we study the Crank-Nicolson method for temporal dimension and the piecewise quadratic polynomial collocation method for spatial dimensions of time-dependent nonlocal problems. The new theoretical results of such discretization are that the proposed numerical method is unconditionally stable and its global truncation error is of $\mathcal{O}\left(\tau^2+h^{4-\gamma}\right)$ with $0<\gamma<1$, where $\tau$ and $h$ are the discretization sizes in the temporal and spatial dimensions respectively. Also we develop the conjugate gradient squared method to solving the resulting discretized nonsymmetric and indefinite systems arising from time-dependent nonlocal problems including two-dimensional cases. By using additive and multiplicative Cauchy kernels in non-local problems, structured coefficient matrix-vector multiplication can be performed efficiently in the conjugate gradient squared iteration. Numerical examples are given to illustrate our theoretical results and demonstrate that the computational cost of the proposed method is of $O(M \log M)$ operations where $M$ is the number of collocation points.

A second-order accurate scheme for two-dimensional space fractional diffusion equations with time Caputo-Fabrizio fractional derivative

We provide and analyze a second order scheme for the model describing the functional distributions of particles performing anomalous motion with exponential Debye pattern and no-time-taking jumps eliminated, and power-law jump length. The equation is derived by continuous time random walk model, being called the space fractional diffusion equation with the time Caputo-Fabrizio fractional derivative. The designed schemes are unconditionally stable and have the second order global truncation error with the nonzero initial condition, being theoretically proved and numerically verified by two methods (a prior estimate with L2-norm and mathematical induction with l∞ norm). Moreover, the optimal estimates are obtained.

Numerical solution of bipolar fuzzy initial value problem

Differential equations occur in many fields of science, engineering and social science as it is a natural way of modeling uncertain dynamical systems. A bipolar fuzzy set model is useful mathematical tool for addressing uncertainty which is an extension of fuzzy set model. In this paper, we study differential equations in bipolar fuzzy environment. We introduce the concept gH-derivative of bipolar fuzzy valued function. We present some properties of gH-differentiability of bipolar fuzzy valued function by considering different types of differentiability. We consider bipolar fuzzy Taylor expansion. By using Taylor expansion, Euler method is presented for solving bipolar fuzzy initial value problems. We discuss convergence analysis of proposed method. We describe some numerical examples to see the convergence and stability of the method and compute global truncation error. From numerical results, we see that for small step size Euler method converges to exact solution.

Development of Improved Scheme for Numerical Integration of Autonomous and Non-Autonomous Initial Value Problems

In this research study, an explicit and single stepnew third order improved scheme for numerical integration of autonomous and non-autonomous initial value problemsalso called Cauchy problems in ordinary differential equations has been developed. Linear stability analysis with corresponding stability region is drawn and error analysis has been provided to confirm third order accuracy of the improved scheme. Inclusion of a partial derivative with respect to the dependent variable with in two slopes of the improved scheme has improved its efficiency in terms of local and global truncation errors. Finally, numerical examples are provided to show performance of the improved scheme in comparison with other existing methods having same order of local accuracy. Throughout the study it has been observed from the results and comparison of the improved scheme that develop algorithm is superior to some existing numerical methods of same order in term of error analysis as well as accuracy viewpoint.The software MATLAB is used to justify the results and graphical representation of the improved scheme.

Automatic, unstructured mesh optimization for simulation and assessment of tide- and surge-driven hydrodynamics in a longitudinal estuary: St. Johns River

A localized truncation error analysis with complex derivatives (LTEA+CD) is applied recursively with advanced circulation (ADCIRC) simulations of tides and storm surge for finite element mesh optimization. Mesh optimization is demonstrated with two iterations of LTEA+CD for tidal simulation in the lower 200 km of the St. Johns River, located in northeast Florida, and achieves more than an over 50% decrease in the number of mesh nodes, relating to a twofold increase in efficiency, at a zero cost to model accuracy. The recursively generated meshes using LTEA+CD lead to successive reductions in the global cumulative truncation error associated with the model mesh. Tides are simulated with root mean square error (RMSE) of 0.09–0.21 m and index of agreement (IA) values generally in the 80s and 90s percentage ranges. Tidal currents are simulated with RMSE of 0.09–0.23 m s−1 and IA values of 97% and greater. Storm tide due to Hurricane Matthew 2016 is simulated with RMSE of 0.09–0.33 m and IA values of 75–96%. Analysis of the LTEA+CD results shows the M2 constituent to dominate the node spacing requirement in the St. Johns River, with the M4 and M6 overtides and the STEADY constituent contributing some. Friction is the predominant physical factor influencing the target element size distribution, especially along the main river stem, while frequency (inertia) and Coriolis (rotation) are supplementary contributing factors. The combination of interior- and boundary-type computational molecules, providing near-full coverage of the model domain, renders LTEA+CD an attractive mesh generation/optimization tool for complex coastal and estuarine domains. The mesh optimization procedure using LTEA+CD is automatic and extensible to other finite element-based numerical models. Discussion is provided on the scope of LTEA+CD, the starting point (mesh) of the procedure, the user-specified scaling of the LTEA+CD results, and the iteration (termination) of LTEA+CD for mesh optimization.

Energy estimates for two-dimensional space-Riesz fractional wave equation

The fractional wave equation governs the propagation of mechanical diffusive waves in viscoelastic media which exhibits a power-law creep, and consequently provided a physical interpretation of this equation in the framework of dynamic viscoelasticity. In this paper, we first develop the energy method to estimate the one-dimensional space-Riesz fractional wave equation. For two-dimensional cases with the variable coefficients, the discretized matrices are proved to be commutative, which ensures to carry out of the priori error estimates. The unconditional stability and convergence with the global truncation error $\mathcal{O}(\tau^2+h^2)$ are theoretically proved and numerically verified. In particulary, the framework of the priori error estimates and convergence analysis are still valid for the compact finite difference scheme and the nonlocal wave equation.

Trajectory errors of different numerical integration schemes diagnosed with the MPTRAC advection module driven by ECMWF operational analyses

Abstract. The accuracy of trajectory calculations performed by Lagrangian particle dispersion models (LPDMs) depends on various factors. The optimization of numerical integration schemes used to solve the trajectory equation helps to maximize the computational efficiency of large-scale LPDM simulations. We analyzed global truncation errors of six explicit integration schemes of the Runge–Kutta family, which we implemented in the Massive-Parallel Trajectory Calculations (MPTRAC) advection module. The simulations were driven by wind fields from operational analysis and forecasts of the European Centre for Medium-Range Weather Forecasts (ECMWF) at T1279L137 spatial resolution and 3 h temporal sampling. We defined separate test cases for 15 distinct regions of the atmosphere, covering the polar regions, the midlatitudes, and the tropics in the free troposphere, in the upper troposphere and lower stratosphere (UT/LS) region, and in the middle stratosphere. In total, more than 5000 different transport simulations were performed, covering the months of January, April, July, and October for the years 2014 and 2015. We quantified the accuracy of the trajectories by calculating transport deviations with respect to reference simulations using a fourth-order Runge–Kutta integration scheme with a sufficiently fine time step. Transport deviations were assessed with respect to error limits based on turbulent diffusion. Independent of the numerical scheme, the global truncation errors vary significantly between the different regions. Horizontal transport deviations in the stratosphere are typically an order of magnitude smaller compared with the free troposphere. We found that the truncation errors of the six numerical schemes fall into three distinct groups, which mostly depend on the numerical order of the scheme. Schemes of the same order differ little in accuracy, but some methods need less computational time, which gives them an advantage in efficiency. The selection of the integration scheme and the appropriate time step should possibly take into account the typical altitude ranges as well as the total length of the simulations to achieve the most efficient simulations. However, trying to summarize, we recommend the third-order Runge–Kutta method with a time step of 170 s or the midpoint scheme with a time step of 100 s for efficient simulations of up to 10 days of simulation time for the specific ECMWF high-resolution data set considered in this study. Purely stratospheric simulations can use significantly larger time steps of 800 and 1100 s for the midpoint scheme and the third-order Runge–Kutta method, respectively.

High Order Algorithm for the Time-Tempered Fractional Feynman–Kac Equation

We provide and analyze the high order algorithms for the model describing the functional distributions of particles performing anomalous motion with power-law jump length and tempered power-law waiting time. The model is derived in [Wu, Deng, and Barkai, Phys. Rev. E., 84 (2016), 032151], being called the time-tempered fractional Feynman-Kac equation. The key step of designing the algorithms is to discretize the time tempered fractional substantial derivative, being defined as $${^S\!}D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!=\!D_t^{\gamma,\widetilde{\lambda}} G(x,p,t)\!-\!\lambda^\gamma G(x,p,t) ~{\rm with}~\widetilde{\lambda}=\lambda+ pU(x),\, p=\rho+J\eta,\, J=\sqrt{-1},$$ where $$D_t^{\gamma,\widetilde{\lambda}} G(x,p,t) =\frac{1}{\Gamma(1-\gamma)} \left[\frac{\partial}{\partial t}+\widetilde{\lambda} \right] \int_{0}^t{\left(t-z\right)^{-\gamma}}e^{-\widetilde{\lambda}\cdot(t-z)}{G(x,p,z)}dz,$$ and $\lambda \ge 0$, $0<\gamma<1$, $\rho>0$, and $\eta$ is a real number. The designed schemes are unconditionally stable and have the global truncation error $\mathcal{O}(\tau^2+h^2)$, being theoretically proved and numerically verified in {\em complex} space. Moreover, some simulations for the distributions of the first passage time are performed, and the second order convergence is also obtained for solving the `physical' equation (without artificial source term).

Reliability of Lyapunov characteristic exponents computed by the two-particle method

For highly complex problems, such as the post-Newtonian formulation of compact binaries, the two-particle method may be a better, or even the only, choice to compute the Lyapunov characteristic exponent (LCE). This method avoids the complex calculations of variational equations compared with the variational method. However, the two-particle method sometimes provides spurious estimates to LCEs. In this paper, we first analyze the equivalence in the definition of LCE between the variational and two-particle methods for Hamiltonian systems. Then, we develop a criterion to determine the reliability of LCEs computed by the two-particle method by considering the magnitude of the initial tangent (or separation) vector ξ0 (or δ0), renormalization time interval τ, machine precision ε, and global truncation error ϵT. The reliable Lyapunov characteristic indicators estimated by the two-particle method form a V-shaped region, which is restricted by d0, ε, and ϵT. Finally, the numerical experiments with the Hénon–Heiles system, the spinning compact binaries, and the post-Newtonian circular restricted three-body problem strongly support the theoretical results.

A second-order accurate numerical method for the space–time tempered fractional diffusion-wave equation

This paper focuses on providing the high order algorithms for the space–time tempered fractional diffusion-wave equation. The designed schemes are unconditionally stable and have the global truncation error O ( τ 2 + h 2 ) , being theoretically proved and numerically verified.

Numerical Analysis of the Mixed 4th-Order Runge-Kutta Scheme of Conditional Nonlinear Optimal Perturbation Approach for the EI Niño-Southern Oscillation Model

AbstractIn this paper, we proposes and analyzes the mixed 4th-order Runge-Kutta scheme of conditional nonlinear perturbation (CNOP) approach for the EI Niño-Southern Oscillation (ENSO) model. This method consists of solving the ENSO model by using a mixed 4th-order Runge-Kutta method. Convergence, the local and global truncation error of this mixed 4th-order Runge-Kutta method are proved. Furthermore, optimal control problem is developed and the gradient of the cost function is determined.

A Low Complexity Algorithm for Non-Monotonically Evolving Fronts

A new algorithm is proposed to describe the propagation of fronts advected in the normal direction with prescribed speed function F. The assumptions on F are that it does not depend on the front itself, but can depend on space and time. Moreover, it can vanish and change sign. The novelty of our method is that its overall computational complexity is predicted to be comparable to that of the Fast Marching Method (Sethian in Proceedings of the National Academy Sciences 93:1591---1595, 1996); (Vladimirsky in Interfaces Free Bound 8(3):281---300, 2006) in most instances. This latter algorithm is $$\mathcal {O}(N^{n}\log N^{n})$$O(NnlogNn) if the computational domain comprises $$N^{n}$$Nn points. We use it in regions where the speed is bounded away from zero--and switch to a different formalism when $$F \approx 0$$Fź0. To this end, a collection of so-called sideways partial differential equations is introduced. Their solutions locally describe the evolving front and depend on both space and time. The well-posedness and geometric properties of those equations are addressed. We then propose a convergent and stable discretization of the PDEs. The resulting algorithm is presented together with a thorough discussion of its features. The accuracy of the scheme is tested when F depends on both space and time. Each example yields an $$\mathcal {O}(1/N)$$O(1/N) global truncation error. We conclude with a discussion of the advantages and limitations of our method.

Discretized fractional substantial calculus

This paper discusses the properties and the numerical discretizations of the fractional substantial integral $$I_s^\nu f(x)=\frac{1}{\Gamma(\nu)} \int_{a}^x{\left(x-\tau\right)^{\nu-1}}e^{-\sigma(x-\tau)}{f(\tau)}d\tau,\nu>0, $$ and the fractional substantial derivative $$D_s^\mu f(x)=D_s^m[I_s^\nu f(x)], \nu=m-\mu,$$ where $D_s=\frac{\partial}{\partial x}+\sigma=D+\sigma$, $\sigma$ can be a constant or a function without related to $x$, say $\sigma(y)$; and $m$ is the smallest integer that exceeds $\mu$. The Fourier transform method and fractional linear multistep method are used to analyze the properties or derive the discretized schemes. And the convergences of the presented discretized schemes with the global truncation error $\mathcal{O}(h^p)$$ (p=1,2,3,4,5)$ are theoretically proved and numerically verified.

Fourth Order Accurate Scheme for the Space Fractional Diffusion Equations

Because of the nonlocal properties of fractional operators, higher order schemes play more important role in discretizing fractional derivatives than classical ones. The striking feature is that higher order schemes of fractional derivatives can keep the same computation cost with first-order schemes but greatly improve the accuracy. Nowadays, there are already two types of second order discretization schemes for space fractional derivatives: the first type is given and discussed in [Sousa & Li, arXiv:1109.2345; Chen & Deng, arXiv:1304.3788; Chen et al., Appl. Numer. Math., 70, 22-41]; and the second type is a class of schemes presented in [Tian et al., arXiv:1201.5949]. The core object of this paper is to derive a class of fourth order approximations, called the weighted and shifted Lubich difference (WSLD) operators, for space fractional derivatives. Then we use the derived schemes to solve the space fractional diffusion equation with variable coefficients in one-dimensional and two-dimensional cases. And the unconditional stability and the convergence with the global truncation error $\mathcal{O}(\tau^2+h^4)$ are theoretically proved and numerically verified.