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The paper is devoted to Fermi-Pasta-Ulam type system that describe an infinite system of nonlinearly coupled particles with nonlocal interaction on a two dimensional integer-valued lattice. It is assumed that each particle interacts nonlinearly with several neighbors horizontally and vertically on both sides. This system forms an infinite system of ordinary differential equations and is representative of a wide class of systems called lattice dynamical systems, which have been extensively studied in recent decades. Among the solutions of such systems, traveling waves deserve special attention. The main result concerns the existence of traveling waves solutions with periodic velocity profiles. Note that the profiles of such waves are not necessarily periodic. The problem of the existence of such solutions is reduced to a variational problem for the action functionals. We obtain sufficient conditions for the existence of such solutions with the aid of the critical point method and the Linking Theorem for functionals satisfying the Palais-Smale condition and possessing linking geometry. We prove that under natural assumptions there exist subsonic traveling waves. While in our previous paper [12], the existence of supersonic periodic traveling waves in this system was established using variational techniques and a corresponding version of the Mountain Pass Theorem for action functionals that satisfy the Cerami condition instead of the Palais-Smale condition. [ABSTRACT FROM AUTHOR]

The paper considers the numerical stability of the backward recurrence algorithm (BRalgorithm) for computing approximants of the continued fraction with complex elements. The new method establishes sufficient conditions for the numerical stability of this algorithm and the error bounds of the calculation of the nth approximant of the continued fraction with complex elements. It follows from the obtained conditions that the numerical stability of the algorithm depends not only on the rounding errors of the elements and errors of machine operations but also on the value sets and the element sets of the continued fraction. The obtained results were used to study the numerical stability of the BR-algorithm for computing the approximants of the continued fraction expansion of the ratio of Horn's confluent functions H7. Bidisc and bicardioid regions are established, which guarantee the numerical stability of the BR-algorithm. The obtained result is applied to the study of the numerical stability of computing approximants of the continued fraction expansion of the ratio of Horn's confluent function H7 with complex parameters. In addition, the analysis of the relative errors arising from the computation of approximants using the backward recurrence algorithm, the forward recurrence algorithm, and Lenz's algorithm is given. The method for studying the numerical stability of the BR-algorithm proposed in the paper can be used to study the numerical stability of the branched continued fraction expansions and numerical branched continued fractions with elements in angular and parabolic domains. [ABSTRACT FROM AUTHOR]

A second‐order accurate, linear numerical method is analyzed for the Landau–Lifshitz equation with large damping parameters. This equation describes the dynamics of magnetization, with a non‐convexity constraint of unit length of the magnetization. The numerical method is based on the second‐order backward differentiation formula in time, combined with an implicit treatment for the linear diffusion term from the harmonic mapping part and explicit extrapolation for the nonlinear terms. Afterward, a projection step is applied to normalize the numerical solution at a point‐wise level. This numerical scheme has shown extensive advantages in the practical computations for the physical model with large damping parameters, which comes from the fact that only a linear system with constant coefficients (independent of both time and the updated magnetization) needs to be solved at each time step, and has greatly improved the numerical efficiency. Meanwhile, a theoretical analysis for this linear numerical scheme has not been available. In this paper, we provide a rigorous error estimate of the numerical scheme, in the discrete ℓ∞(0,T;ℓ2)∩ℓ2(0,T;Hh1)$$ {\ell}^{\infty}\left(0,T;{\ell}^2\right)\cap {\ell}^2\left(0,T;{H}_h^1\right) $$ norm, under suitable regularity assumptions and reasonable ratio between the time step size and the spatial mesh size. In particular, the projection operation is nonlinear, and a stability estimate for the projection step turns out to be highly challenging. Such a stability estimate is derived in details, which will play an essential role in the convergence analysis for the numerical scheme, if the damping parameter is greater than 3. [ABSTRACT FROM AUTHOR]

In this work we study an inverse problem of recovering a space-time-dependent diffusion coefficient in the subdiffusion model from the distributed observation, where the mathematical model involves a Djrbashian–Caputo fractional derivative of order |$\alpha \in (0,1)$| in time. The main technical challenges of both theoretical and numerical analyses lie in the limited smoothing properties due to the fractional differential operator and high degree of nonlinearity of the forward map from the unknown diffusion coefficient to the distributed observation. We establish two conditional stability results using a novel test function, which leads to a stability bound in |$L^2(0,T;L^2(\varOmega))$| under a suitable positivity condition. The positivity condition is verified for a large class of problem data. Numerically, we develop a rigorous procedure for recovering the diffusion coefficient based on a regularized least-squares formulation, which is then discretized by the standard Galerkin method with continuous piecewise linear elements in space and backward Euler convolution quadrature in time. We provide a complete error analysis of the fully discrete formulation, by combining several new error estimates for the direct problem (optimal in terms of data regularity), a discrete version of fractional maximal |$L^p$| regularity and a nonstandard energy argument. Under the positivity condition, we obtain a standard |$\ell ^2(L^2(\varOmega))$| error estimate consistent with the conditional stability. Further, we illustrate the analysis with some numerical examples. [ABSTRACT FROM AUTHOR]

In this paper, we present an enriched Ciarlet-Raviart scheme for the biharmonic equation with variable coefficient on Lipschitz (maybe nonconvex)polyhedral domains. With the enriched finite element space for the Laplacian of the true solution, we manage to prove the discrete $ H^{2} $-stability of numerical solution. Error analysis is provided for solutions with low regularity. [ABSTRACT FROM AUTHOR]

We consider a predictor–corrector numerical method for solving Caputo–Hadamard fractional differential equation over the uniform mesh log t j = log a + log t N a j N , j = 0 , 1 , 2 , ... , N with a ≥ 1 , where log a = log t 0 < log t 1 < ... < log t N = log T is a partition of [ log a , log T ] . The error estimates under the different smoothness properties of the solution y and the nonlinear function f are studied. Numerical examples are given to verify that the numerical results are consistent with the theoretical results. [ABSTRACT FROM AUTHOR]

Hyperbolic partial differential equations are frequently referenced in modeling real-world problems in mathematics and engineering. Therefore, in this study an initial-boundary value issue is proposed for the pseudo-hyperbolic telegraph equation. By operator method, converting the PDE to an ODE provides an exact answer to this problem. After that, finite difference method is applied to construct first order finite difference schemes to calculate approximate numerical solutions. The stability estimations of finite difference schemes are shown, as well as some numerical tests to check the correctness in comparison to the precise solution. The numerical solution is subjected to error analysis. As a result of the error analysis, the maximum norm errors tend to decrease as we increase the grid points. It can be drawn that the established scheme is accurate and effective. [ABSTRACT FROM AUTHOR]

In this paper, a class of second-order singularly perturbed interior layer problems is examined. A nonpolynomial mixed spline is used to develop the tridiagonal scheme. The developed method is second as well as fourthorder accurate based on the parameters. Error analysis is also carried out. The method is shown to converge point-wise to the true solution with higher accuracy. Linear and nonlinear second-order singularly perturbed boundary value problems have been solved by the presented method. Five numerical illustrations are given to demonstrate the applicability of the proposed method. Absolute errors are given in tables, which show that our method is more efficient than previously existing methods. [ABSTRACT FROM AUTHOR]

In incompressible flow problems, the finite element discretization of pressure and velocity can be done through either stable spaces or stabilized pairs. For equal-order stabilized methods with piecewise linear discretization, the classical theory guarantees only linear convergence for the pressure approximation. However, a higher order is often observed, yet seldom discussed, in numerical practice. Such experimental observations may, in the absence of a sound a priori error analysis, mislead the selection of finite element spaces in practical applications. Therefore, we present here a numerical analysis demonstrating that an initial higher-order pressure convergence may in fact occur under certain conditions, for equal-order elements of any degree. Moreover, our numerical experiments clearly indicate that whether and for how long this behavior holds is a problem-dependent matter. These findings confirm that an optimal pressure convergence can in general not be expected when using unbalanced velocity-pressure pairs. [ABSTRACT FROM AUTHOR]

In this paper, a numerical method based on modified hat functions (MHFs) is investigated to find an approximate solution for a fractional-order system of stochastic integro-differential equations. The fractional and stochastic operational matrices of integration of these functions are employed to present the numerical approach. By using these operational matrices and properties of MHFs, the considered problem is transformed into a system of algebraic equations which can be easily solved by an iterative method. Also, error analysis of the proposed method is discussed. At the end, the accuracy and effectiveness of this approach are studied by some numerical examples. [ABSTRACT FROM AUTHOR]

As a typical artifact in machining, the roundness of shaft artifacts is an important tolerance standard. How to measure with high precision in ultra-precision machining is a difficult point. At present, the three-probe method is the most widely used method to measure spindle rotation error and artifact roundness error, which has the advantage of on-machine measurement, but the three-probe method has the critical problem of harmonic suppression. To completely separate the spindle error when measuring the roundness of the artifact, a new virtual parallel two-point measurement system is proposed in this paper. The probe is fixed on the flexible hinge stage, which is driven by a voice coil motor so that the measurement function of two probes can be realized only by using a single probe. Detailed numerical analysis and simulation analysis have been carried out to verify that the method can completely separate the spindle error in theory. Finally, the accuracy of the measurement system is tested through experiments. The new method results are compared with the single-point scanning method without error separation and the high-precision roundness instrument Talyrond 585LT. The comparison results show that the new method can separate the rotation error of the spindle and can improve the roundness measurement accuracy. [ABSTRACT FROM AUTHOR]

Evaluating the log-sum-exp function or the softmax function is a key step in many modern data science algorithms, notably in inference and classification. Because of the exponentials that these functions contain, the evaluation is prone to overflow and underflow, especially in low-precision arithmetic. Software implementations commonly use alternative formulas that avoid overflow and reduce the chance of harmful underflow, employing a shift or another rewriting. Although mathematically equivalent, these variants behave differently in floating-point arithmetic and shifting can introduce subtractive cancellation. We give rounding error analyses of different evaluation algorithms and interpret the error bounds using condition numbers for the functions. We conclude, based on the analysis and numerical experiments, that the shifted formulas are of similar accuracy to the unshifted ones, so can safely be used, but that a division-free variant of softmax can suffer from loss of accuracy. [ABSTRACT FROM AUTHOR]

In this paper, error analysis of the unstructured mesh Galerkin finite element method for the two-dimensional time-space fractional Schrödinger equation with a time-independent potential defined on a finite domain is studied. The finite difference method is used to discretize the Caputo time fractional derivative, while the finite element method using unstructured mesh is used to deal with the Riesz fractional operators in space. Both the stability and convergence analysis of the numerical scheme are constructed. Numerical example is conducted to testify the validity of the proposed method. The conservation of the space fractional Schrödinger equation and the non-conservation of the time fractional Schrödinger equation in quantum mechanical system are achieved. This paper proposes an efficient numerical method as well as its theoretical analysis for the two-dimensional time-space fractional Schrödinger equation with time-independent potentials. [ABSTRACT FROM AUTHOR]

This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected L² error of the method can be estimated, up to logarithmic factors, by H + (ε/H)d/2, ε being the small correlation length of the random coefficient and H the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization. [ABSTRACT FROM AUTHOR]

In this article, we analyse an integral equation of the second kind that represents the solution of N interacting dielectric spherical particles undergoing mutual polarisation. A traditional analysis can not quantify the scaling of the stability constants- and thus the approximation error- with respect to the number N of involved dielectric spheres. We develop a new a priori error analysis that demonstrates N-independent stability of the continuous and discrete formulations of the integral equation. Consequently, we obtain convergence rates that are independent of N. [ABSTRACT FROM AUTHOR]

In this paper, a number of energy stable numerical schemes are proposed for the fractional Cahn–Hilliard equation. We prove mass conservation, unique solvability and energy stability for three time semi-discretized schemes based on the first-order semi-implicit scheme, the Crank–Nicolson scheme and the BDF2 scheme respectively. Then we present error analysis for these numerical schemes with the Fourier spectral approximation in space. Some numerical experiments are finally carried out to confirm accuracy and effectiveness of these proposed methods. [ABSTRACT FROM AUTHOR]

In this paper we provide key estimates used in the stability and error analysis of discontinuous Galerkin finite element methods (DGFEMs) on domains with curved boundaries. In particular, we review trace estimates, inverse estimates, discrete Poincaré–Friedrichs' inequalities, and optimal interpolation estimates in noninteger Hilbert–Sobolev norms, that are well known in the case of polytopal domains. We also prove curvature bounds for curved simplices, which does not seem to be present in the existing literature, even in the polytopal setting, since polytopal domains have piecewise zero curvature. We demonstrate the value of these estimates, by analyzing the IPDG method for the Poisson problem, introduced by Douglas and Dupont, and by analyzing a variant of the hp‐DGFEM for the biharmonic problem introduced by Mozolevski and Süli. In both cases we prove stability estimates and optimal a priori error estimates. Numerical results are provided, validating the proven error estimates. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop and analyze numerical schemes for approximately solving the backward problem of subdiffusion equation involving a fractional derivative in time with order α ∈ (0, 1). After using quasi-boundary value method to regularize the 'mildly' ill-posed problem, we propose a fully discrete scheme by applying finite element method (FEM) in space and convolution quadrature (CQ) in time. We provide a thorough error analysis of the resulting discrete system in both cases of smooth and nonsmooth data. The analysis relies heavily on smoothing properties of (discrete) solution operators, and nonstandard error estimate for the direct problem in terms of problem data regularity. The theoretical results are useful to balance discretization parameters, regularization parameter and noise level. Numerical examples are presented to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we deal with the numerical analysis of the Lord–Shulman thermoelastic problem with porosity and microtemperatures. The thermomechanical problem leads to a coupled system composed of linear hyperbolic partial differential equations written in terms of transformations of the displacement field and the volume fraction, the temperature and the microtemperatures. An existence and uniqueness result is stated. Then, a fully discrete approximation is introduced using the finite element method and the implicit Euler scheme. A discrete stability property is shown, and an a priori error analysis is provided, from which the linear convergence is derived under suitable regularity conditions. Finally, some numerical simulations are presented to demonstrate the accuracy of the approximation, the comparison with the classical Fourier theory and the behavior of the solution in two-dimensional examples. [ABSTRACT FROM AUTHOR]

The discounted price process under the constant elasticity of variance (CEV) model is not a martingale (wrt the risk-neutral measure) for options markets with upward sloping implied volatility smiles. The loss of the martingale property implies the existence of (at least) two option prices for the call option: the price for which the put-call parity holds and the (risk-neutral) price representing the lowest cost of replicating the call payoff. This article derives closed-form solutions for the Greeks of the risk-neutral call option pricing solution that are valid for any CEV process exhibiting forward skew volatility smile patterns. Using an extensive numerical analysis, we conclude that the differences between the call prices and Greeks of both solutions are substantial, which might yield significant errors of analysis for pricing and hedging purposes. [ABSTRACT FROM AUTHOR]

This work aims at employing the discontinuous Galerkin (DG) methods for solving the nonstationary conduction–convection model, which is motivated by advantages of DG methods. The discontinuous element pair P k - P k − 1 - P k in the space discretization and backward Euler scheme in time are applied, as well as an explicit upwind scheme based on Picard iteration is introduced for the coupled term. Existence and uniqueness of the approximation solutions are proved, along with preserving the unconditional stability of the numerical schemes. The priori error estimates for velocity, pressure and temperature are also derived. Numerical tests are provided to verify the theoretical analysis and to indicate the efficiency of the DG methods in simulating practical problems. [ABSTRACT FROM AUTHOR]

Standard backward error analyses for numerical linear algebra algorithms provide worst-case bounds that can significantly overestimate the backward error. Our recent probabilistic error analysis, which assumes rounding errors to be independent random variables [SIAM J. Sci. Comput., 41 (2019), pp. A2815-A2835], contains smaller constants but its bounds can still be pessimistic. We perform a new probabilistic error analysis that assumes both the data and the rounding errors to be random variables and assumes only mean independence. We prove that for data with zero or small mean we can relax the existing probabilistic bounds of order √nu to much sharper bounds of order u, which are independent of n. Our fundamental result is for summation and we use it to derive results for inner products, matrix-vector products, and matrix-matrix products. The analysis answers the open question of why random data distributed on [-1, 1] leads to smaller error growth for these kernels than random data distributed on [0, 1]. We also propose a new algorithm for multiplying two matrices that transforms the rows of the first matrix to have zero mean and we show that it can achieve significantly more accurate results than standard matrix multiplication. [ABSTRACT FROM AUTHOR]

Electromagnetic (EM) waves do not travel in straight lines in the troposphere due to refraction. This leads to errors in many engineering applications. In order to correct these errors, a numerical solution based on the difference method has been proposed, which can reveal the degree of EM wave bending owing to atmospheric refraction, moreover, it can predict the actual path of the EM wave propagating in the troposphere. The error of the numerical method is studied here, which reveals the source of the error. Based on this, two improvement methods are proposed. The computational accuracy of the numerical methods is verified by comparison with the analytical solution. The proposed method ensures the computational efficiency and accuracy of the problem of refraction of EM waves in the troposphere. [ABSTRACT FROM AUTHOR]

We consider a one dimensional elliptic distributed optimal control problem with pointwise constraints on the derivative of the state. By exploiting the variational inequality satisfied by the derivative of the optimal state, we obtain higher regularity for the optimal state under appropriate assumptions on the data. We also solve the optimal control problem as a fourth order variational inequality by a C1 finite element method, and present the error analysis together with numerical results. [ABSTRACT FROM AUTHOR]

Heat transfer coefficient (HTC) is a useful concept to characterize the heat exchange performance of rock fracture. However, existing study has preliminarily found that some HTC formulas would lead to abnormal values, which claims a systematical evaluation of existing formulas. In this paper, 214 test records are employed to evaluate 8 existing formulas from literature. It is found that, for all these experimental data, Bai's formula (Formula D) developed in 2017 shows fairly good numerical stability, while the rest show numerical oscillations and anomalies in different degrees. Moreover, the numerical oscillations and anomalies would be exacerbated with the increase of flow rates, which leads to poor applicability. Error propagation theory in numerical analysis is used to analyze the mechanism of the numerical oscillations and anomalies which can be effectively explained with the condition number and the relative error of the corresponding formula. [ABSTRACT FROM AUTHOR]

Anisotropic elliptic interface problems are important but hard to solve either analytically or numerically. There is limited literature on numerical methods based on structured meshes. Finite element methods are often used, but the usual average error estimates cannot guarantee accuracy of the solution near or at the interface. For finite difference methods, it is challenging to discretize mixed derivatives and carry out the convergence analysis. In this paper, a new finite element-finite difference (FE-FD) method that combines a finite element discretization (away from the interface) whose coefficient matrix is a symmetric semipositive definite, with a finite difference discretization (near or on the interface) whose coefficient matrix part has properties of an M-matrix, is developed. An interpolation scheme based on the immersed interface method is also applied to compute the normal derivative of solution (or gradient) accurately from each side of the interface. Error analysis and numerical experiments are also presented. [ABSTRACT FROM AUTHOR]

In this paper, the existence and uniqueness of solution to Caputo–Hadamard fractional differential equation (FDE) are studied. The continuation theorem is established too. Then, Euler and predictor–corrector methods are built up to solve Caputo–Hadamard FDE. The stability and error analysis of the derived numerical schemes are investigated as well. At last, a numerical example is carried out to verify the numerical algorithm. [ABSTRACT FROM AUTHOR]

In this work, we study a new spectral Petrov-Galerkin approximation of space-time fractional reaction-diffusion equations with viscosity terms built by Riemann-Liouville fractional-order derivatives. The proposed method is reliant on generalized Jacobi functions (GJFs) for our problems. The contributions are threefold: First, thanks to the theoretical framework of variational problems, the well-posedness of the problem is proved. Second, new GJF-basis functions are established to fit weak solutions, which take full advantages of the global properties of fractional derivatives. Moreover, the basis functions conclude singular terms, in order to solve our problems with given smooth source term. Finally, we get a numerical analysis of error estimates to depend on GJF-basis functions. Numerical experiments confirm the expected convergence. In addition, they are given to show the effect of the viscosity terms in anomalous diffusion. [ABSTRACT FROM AUTHOR]

Boundary-value problems involving the linear differential equation εy" - xy' + y = 0 have surprising properties as ε → 0. We examine this equation from eight points of view, showing how it sheds light on aspects of numerical analysis (backward error analysis and illconditioning), asymptotics (boundary layer analysis), dynamical systems (slow manifolds), ODE theory (Sturm--Liouville operators), spectral theory (eigenvalues and pseudospectra), sensitivity analysis (adjoints and SVD), physics (ghost solutions), and PDE theory (Lewy nonexistence). [ABSTRACT FROM AUTHOR]

For pixel-based microstructure representations we propose two different variants of adaptive, quadtree-based mesh coarsening algorithms that serve the purpose of a preprocessor for finite element analyses in the context of numerical homogenization. High resolution is preserved at interfaces for accuracy, coarse-graining in the defect-free interior of phases for efficiency. Error analysis is carried out on the micro scale by error estimation which itself is assessed by true error computation. Modified stress recovery schemes for an error indicator are proposed which overcome the deficits of the standard superconvergent recovery scheme for nodal stress computation in cases of interfaces with stiffness jump. By virtue of error analysis the improved efficiency by the reduction of unknowns is put into relation to the increase of the discretization error and thereby sets a rational basis for decisions on favorable meshes having the best trade-off between accuracy and efficiency as underpinned by various examples. [ABSTRACT FROM AUTHOR]

Micro lubrication of a gas between two walls with an arbitrary temperature difference is studied on the basis of the Bhatnagar–Gross–Krook–Welander model of the Boltzmann equation. Applying the slowly varying approximation, the kinetic equation is studied analytically when the Knudsen number based on the gap size is large. The leading order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. Due to this crude approximation, a macroscopic lubrication model of Reynolds-type equation is derived in a closed form. For an assessment of the model, a direct numerical analysis of the kinetic equation is also conducted. The lift calculated using our model approximates that of the direct numerical analysis within the error of 4% uniformly in the range of the temperature ratio between 0.75 and 2 and the Knudsen number Kn between 0.1 and 10. A heating of the moving wall reduces the lift acting on the other wall when Kn is sufficiently large, whereas it is enhanced when Kn is sufficiently small. [ABSTRACT FROM AUTHOR]

In this paper, we address the problem of counting integer points in a rational polytope described by P(y) = {x ∈ Rm: Ax= y, x ≥ 0}, where A is an n × m integer matrix and y is an n-dimensional integer vector. We study the Z transformation approach initiated in works by Brion and Vergne, Beck, and Lasserre and Zeron from the numerical analysis point of view and obtain a new algorithm on this problem. If A is nonnegative, then the number of integer points in P(y) can be computed in O (poly (n , m , ‖ y ‖ ∞) (‖ y ‖ ∞ + 1) n) time and O (poly (n , m , ‖ y ‖ ∞)) space. This improves, in terms of space complexity, a naive DP algorithm with O ( (‖ y ‖ ∞ + 1) n) -size dynamic programming table. Our result is based on the standard error analysis of the numerical contour integration for the inverse Z transform and establishes a new type of an inclusion-exclusion formula for integer points in P(y). We apply our result to hypergraph b-matching and obtain a O (poly (n , m , ‖ b ‖ ∞) (‖ b ‖ ∞ + 1) (1 − 1 / k) n ) time algorithm for counting b-matchings in a k-partite hypergraph with n vertices and m hyperedges. This result is viewed as a b-matching generalization of the classical result by Ryser for k = 2 and its multipartite extension by Björklund and Husfeldt. [ABSTRACT FROM AUTHOR]

We describe and analyze a streamline-upwinded, Petrov–Galerkin (SUPG) method for numerical simulation of transport models with adsorption kinetics. Specifically, we rewrite the adsorption term to pose the problem using a single mass accumulation term for the liquid phase concentration. The rate of adsorption is used in the nonlinear coefficient for the mass accumulation. We use streamline upwinding to stabilize the advection-dominated applications we consider. We provide a priori error analysis verified by numerical results, and we provide additional numerical results demonstrating the usefulness of our scheme. We consider a variety of adsorption model parameters and describe the performance of our algorithm in these cases. [ABSTRACT FROM AUTHOR]

Although less accurate in comparison with most of the existing high-order panel methods, the Class Hess Smith panel method is still popular for potential flow calculations due to its simplicity, robustness, and especially efficiency. In the application where the method is adopted for the problems of concern, the method is usually first validated and analyzed for numerical accuracy prior to the application. This is a common practice as also seen in almost all the literature. However, the objects chosen for validation and accuracy analysis are often geometrically symmetric (most typically, spheres) and discretized into symmetric meshes. However, such analysis with symmetric meshes does not actually reveal the numerical errors as the body involved in the problem in concern or the flow around the body may not be symmetrical. In this paper, the numerical errors of the Hess Smith method are systematically analyzed with asymmetrical meshes. The values of various quantities associated with isolated panels and also the entire body are obtained and compared with the Hess Smith panel method does exhibit a very different error pattern. [ABSTRACT FROM AUTHOR]

In this work, we consider a new fractional derivative with nonsingular kernel introduced by Caputo-Fabrizio (CF) and propose a finite difference method for computing the CF fractional derivatives. Based on an iterative technique, we can reduce the computational complexity from O(J²N) to O(JN), and the corresponding storage will be cut down from O(JN) to O(N), which makes the computation much more efficient. Besides, by adopting piece-wise Lagrange polynomials of degrees 1, 2, and 3, we derive the second, third, and fourth order discretization formulas respectively. The error analysis and numerical experiments are carefully provided for the validation of the accuracy and efficiency of the presented method. [ABSTRACT FROM AUTHOR]

Studying & understanding the bursting dynamics of membrane potential in neurobiology is captivating in applied sciences, with many features still to be uncovered. In this study, 2D and 3D neuronal activities given by models of Hindmarsh-Rose (HR) neurons with external current input are analyzed numerically with Haar wavelet method, proven to be convergent through error analysis. Our numerical analysis considers two control parameters: the external current Iext and the derivative order γ, on top of the other seven usual parameters a, b, c, d, ν1, ν2 and xrest. Bifurcation scenarios for the model show existence of equilibria, both stable and unstable of type saddle and spiral. They also reveal existence of stable limit cycle toward which the trajectories get closer. Numerical approximations of solutions to the 2D model reveals that equilibria remains the same in all cases, irrespective of control parameters'values, but the observed repeated sequences of impulses increase as γ decreases. This inverse proportionability reveals a system likely to be γcontrolled with γ varying from 1 down to 0. A similar observation is done for the 3D HR neuron model where regular burst is observed and turns into period-adding chaotic bifurcation (burst with uncountable peaks) as γ changes from 1 down to 0. [ABSTRACT FROM AUTHOR]

The electron dynamics with complex third-order Suzuki-Trotter propagator (ST3) has been implemented into a planewave (PW) based density functional theory program, and several applications including linear absorption spectra and coupled electron-nucleus dynamics have been calculated. Since the ST3 reduces the number of Fourier transforms to less than half compared to the fourth-order Suzuki-Trotter propagator (ST4), more than twice faster calculations are possible by exploiting the ST3. We analyzed numerical errors of both the ST3 and the ST4 in the presence/absence of an external field for several molecules such as Al2, N2, and C2H4. We obtained that the ST3 gives the same order of numerical errors (10-5 Ry after 100 fs) as the ST4. Also, the time evolution of dipole moments, hence the absorption spectrum, is equivalent for both ST3 and ST4. As applications, the linear absorption spectrum for an ethylene molecule was studied. From the density difference analysis, we showed that the absorption peaks at 6.10 eV and 7.65 eV correspond to the π → 4ag and π → π* excitation bands, respectively. We also investigated the molecular vibrational effect to the absorption spectra of an ethylene molecule and the dynamics of a hydrogen molecule after the σ → σ* transition by formulating coupled electron-nucleus dynamics within the Ehrenfest regime. The trajectory of nuclei follows the excited state potential energy curve exactly. [ABSTRACT FROM AUTHOR]

The complex scaling operator exp(-θx⁁p⁁/h), being a foundation of the complex scaling method for resonances, is studied in the Wigner phase-space representation. It is shown that the complex scaling operator behaves similarly to the squeezing operator, rotating and amplifying Wigner quasi-probability distributions of the respective wavefunctions. It is disclosed that the distorting effect of the complex scaling transformation is correlated with increased numerical errors of computed resonance energies and widths. The behavior of the numerical error is demonstrated for a computation of CO2 + vibronic resonances. [ABSTRACT FROM AUTHOR]

Trotter's theorem forms the theoretical basis of most modern molecular dynamics. In essence this theorem states that a time displacement operator (a Lie operator) constructed by exponentiating a sum of noncommuting operators can be approximated by a product of single operators provided the time interval is 'very small.' In theory 'very small' implies infinitesimally small (at which point the approximate product becomes exact), while in practical analysis a finite time interval is divided into several small subintervals or steps. It follows, therefore, that the larger the number of steps the better the approximation to the exact time displacement operator. The question therefore arises: How many steps are sufficient? For bounded operators, standard theorems are available to provide the answer. In this paper we show that a very simple combinatorial formula can be derived which allows the computation of the global differences (as a function of the number of steps) between the Taylor coefficients of the exact time displacement operator and an approximate one constructed by using a finite number of steps. The formula holds for both bounded and nonbounded operators and shows, quantitatively, what is qualitatively expected-that the error decreases with increasing number of steps. Furthermore, the formula applies irrespective of the complexity of the system, boundary conditions, or the thermodynamic ensemble employed for averaging the initial conditions. The analysis yields explicit expressions for the Taylor coefficients which are then used to compute the errors. In the case of the algorithmically based practical numerical simulations in which fixed, albeit small, steps are repeatedly applied, the rise in the number of steps does not reduce the size of the steps but increases the total time of interest. The combinatorial formula shows that, here, the errors diverge. Furthermore, this work can be used to supplement other efforts such as the use of shadow Hamiltonians where the truncation of the series expansion of the latter will produce errors in the higher order propagator moments. [ABSTRACT FROM AUTHOR]

We describe perturbative methods for improving finite-basis Hartree–Fock calculations toward the complete-basis limit. The best method appears to offer quadratic error reduction and preliminary numerical applications demonstrate that remarkably accurate Hartree–Fock energies can be obtained. [ABSTRACT FROM AUTHOR]

A main goal of single-molecule experiments is to evaluate equilibrium free energy differences by applying fluctuation relations to repeated work measurements along irreversible processes. We quantify the error that is made in a free energy estimate by means of the Jarzynski equality when the accumulated work expended on the whole system (including the instrument) is erroneously replaced by the work transferred to the subsystem consisting of the sole molecular construct. We find that the error may be as large as 100%, depending on the number of experiments and on the bandwidth of the data acquisition apparatus. Our theoretical estimate is validated by numerical simulations and pulling experiments on DNA hairpins using optical tweezers. [ABSTRACT FROM AUTHOR]

We compute the adiabatic transition energies of methylene (CH2) from the ground state to the lowest electronically excited valence states using the r12-MR-ACPF-2 method with a large basis set and an extended reference space. We recall that this method aims at reaching the basis-set and full configuration interaction (CI) limits simultaneously. Our best excitation energies, Te (T0), are 9.22 (8.87) (ã 1A1, corrected for relativistic and adiabatic effects), 31.98 (31.86) (b 1B1), and 57.62 (57.18) kcal mol-1 (c 1A1) (both uncorrected). We are able to reach the respective basis-set limits that closely that the remaining errors of our (uncorrected) calculations are clearly due to the MR-ACPF-2 method. While we are unable to assess the error of the latter method in a systematic way, we still believe that it is rather unlikely that the errors of our excitation energies exceed ±0.10 kcal mol-1. We finally observe that our (corrected) ã state values deviate by only -0.10 (-0.10) kcal mol-1 from the results of Császár et al. [J. Chem. Phys. 118, 10631 (2003)] — who did careful extrapolations to the valence full-CI and basis-set limits and added a correction for the core correlation — and that the deviation from experiment is only -0.13 (-0.13) kcal mol-1. From these excellent agreements we conclude that our excitation energies to the b and c states are similarly accurate. [ABSTRACT FROM AUTHOR]

We compute the adiabatic transition energies of methylene (CH2) from the ground state to the lowest electronically excited valence states using the r12-MR-ACPF-2 method with a large basis set and an extended reference space. We recall that this method aims at reaching the basis-set and full configuration interaction (CI) limits simultaneously. Our best excitation energies, Te (T0), are 9.22 (8.87) (ã 1A1, corrected for relativistic and adiabatic effects), 31.98 (31.86) (b 1B1), and 57.62 (57.18) kcal mol-1 (c 1A1) (both uncorrected). We are able to reach the respective basis-set limits that closely that the remaining errors of our (uncorrected) calculations are clearly due to the MR-ACPF-2 method. While we are unable to assess the error of the latter method in a systematic way, we still believe that it is rather unlikely that the errors of our excitation energies exceed ±0.10 kcal mol-1. We finally observe that our (corrected) ã state values deviate by only -0.10 (-0.10) kcal mol-1 from the results of Császár et al. [J. Chem. Phys. 118, 10631 (2003)] — who did careful extrapolations to the valence full-CI and basis-set limits and added a correction for the core correlation — and that the deviation from experiment is only -0.13 (-0.13) kcal mol-1. From these excellent agreements we conclude that our excitation energies to the b and c states are similarly accurate. [ABSTRACT FROM AUTHOR]

Stoney's equation and subsequent modifications and some approximations are widely used to evaluate the macrostress within a film on a substrate, though some of these solutions are only applicable for thin films. The purpose of this paper is to review the considerable efforts devoted to the analysis of residual stresses in a single-layer film in the last century and recent years and to estimate the errors involved in using these formulas. The following are some of the important results that can be obtained. (1) The exact solution for the residual stress can be expressed in terms of Stoney's equation [Proc. R. Soc. London A82, 172 (1909)] and a correction factor, (1+Ση3)/(1+η), where Σ,η are the ratios of the elastic modulus and the thickness of the film to those of the substrate, respectively. (2) When the thickness ratio of the film and the substrate is less than 0.1, Stoney's equation and Röll's approximation [J. Appl. Phys. 47, 3224 (1976)] do not cause serious errors. (3) The approximation proposed by Vilms and Kerps [J. Appl. Phys. 53, 1536 (1982)] is an improved modification for Stoney's equation and can be applicable when η≤0.3. (4) The approximations proposed by Brenner and Senderoff [J. Res. Natl. Bur. Stand. 42, 105 (1949)] and Teixeira [Thin Solid Films 392, 276 (2001)] can lead to serious errors and should be avoided. (5) The approximation based on the assumption of constant elastic modulus is only applicable for a ratio of η≤0.01 and can be very misleading. [ABSTRACT FROM AUTHOR]

We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection–diffusion–reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals. [ABSTRACT FROM AUTHOR]

In this paper, the numerical solution of a nonlinear first-order singularly perturbed differential equation with integral boundary condition is considered. The discrete method is generated by a backward Euler formula and the grid is obtained by equidistributing a monitor function based on arc-length. We first give a rigorous error analysis for the numerical method of this problem on a grid that is constructed adaptively from a knowledge of the exact solution. A first-order rate of convergence, independent of the perturbation parameter, is established. Then, an a posteriori error bound and the corresponding convergence result are derived for the presented numerical scheme on an adaptive grid, which is constructed adaptively from a discrete approximation of the exact solution. At last, numerical experiments are given to illustrate our theoretical results. [ABSTRACT FROM AUTHOR]

We aim to present two algorithms for the non-stationary Stokes/Darcy model. The first one is the standard grad–div stabilization scheme. The other one is a modular grad–div based on the standard Backward Euler code which does not crash or slow down for large value grad–div parameters. Both algorithms cannot only improve the efficiency and accuracy of calculation but also can improve mass conservation, while the modular algorithm can be better. We give a complete theoretical analysis of the stability and error estimations of the algorithms. Finally, the theoretical results are verified by numerical experiments and the advantages of adding grad–div stabilization terms are demonstrated. [ABSTRACT FROM AUTHOR]

We discuss the stability theory and numerical analysis of the Helmholtz equation with variable and possibly nonsmooth or oscillatory coefficients. Using the unique continuation principle and the Fredholm alternative, we first give an existence-uniqueness result for this problem, which holds under rather general conditions on the coefficients and on the domain. Under additional assumptions, we derive estimates for the stability constant (i.e., the norm of the solution operator) in terms of the data (i.e., PDE coefficients and frequency), and we apply these estimates to obtain a new finite element error analysis for the Helmholtz equation which is valid at a high frequency and with variable wave speed. The central role played by the stability constant in this theory leads us to investigate its behaviour with respect to coefficient variation in detail. We give, via a 1D analysis, an a priori bound with the stability constant growing exponentially in the variance of the coefficients (wave speed and/or diffusion coefficient). Then, by means of a family of analytic examples (supplemented by numerical experiments), we show that this estimate is sharp. [ABSTRACT FROM AUTHOR]