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A second-order Crank-Nicolson finite difference method is designed to solve a 1D nonlocal Schrödinger equation on the whole real axis. We employ an asymptotically compatible scheme to discretize the spatially nonlocal operator, and apply the Crank-Nicolson scheme in time to achieve a fully discrete infinite system. An iterative technique for the second-order matrix difference equation is then developed to obtain Dirichlet-to-Dirichlet (DtD)-type artificial boundary conditions (ABCs) with the application of z z -transform for the resulting fully discrete system. After that, with the aid of discrete nonlocal Green’s first identity, we derive Dirichlet-to-Neumann (DtN)-type ABCs from DtD-type ABCs. The resulting DtN-type ABCs are available to reduce the infinite discrete system to a finite discrete system on a truncated computational domain, and make it possible to perform stability and convergence analysis for the reduced problem. Finally, numerical examples are provided to demonstrate the effectiveness of the proposed approach.

This paper discusses the application of local interpolation splines of the second order of approximation for the numerical solution of Volterra integral equations of the second kind. Computational schemes based on the use of polynomial and non-polynomial splines are constructed. The advantages of the proposed method include the ability to calculate the integrals which are present in the computational methods. The application of splines to the solution of nonlinear Volterra integral equations is also discussed. The results of numerical experiments are presented

In this paper, a stationary one-dimensional Poisson-Nernst-Planck model with permanent charge is studied under the assumption that n - 1 positively charged ion species have the same valence and the permanent charge is small. By expanding the singular solutions of Poisson-Nernst-Planck model with respect to small permanent charge, the explicit formulae for the zeroth order approximation and the first order approximation of individual flux can be obtained. Based on these explicit formulae, the effects of small permanent charges on individual flux are investigated.

The covariance matrix adaptation evolution strategy (CMA-ES) is one of the most successful evolutionary algorithms. CMA-ES incrementally learns the variable metric by evolving a full covariance matrix. Yet, it suffers from high computational overload. In this paper, we propose two efficient variants of CMA-ES, termed mutation matrix adaptation (MMA-ES) and exponential MMA-ES (xMMA-ES). These variants are derived by taking the first order approximation of the update of the covariance matrix in CMA-ES. Both variants avoid the computational costly matrix decomposition while keeping the simplicity of the update scheme of CMA-ES. We analyze the properties and connections of MMA-ES and xMMA-ES to other variants of evolution strategies. We have experimentally studied the proposed algorithms’ behaviors and performances. xMMA-ES and MMA-ES generally outperform or perform competitively to CMA-ES. We have investigated the performance of MMA-ES with the BiPop restart strategy on the BBOB benchmarks. The experimental results validate the performance of the proposed algorithms.

The problem of testing the scale parameter (γ) of the power-law process for a single system is taken into account when dealing with incomplete failure data. In this study, the modified signed log-l...

In this study, a new method called Second-order Consistent Exponential Approximation (SCEA) is developed to generate the high-quality nonlinear approximation of the structural problems. For this pu...

A special point on each time interval is found for the approximation of the variable-order time Caputo derivative, which makes at least second order approximation accuracy be obtained. On this basis, two difference schemes are proposed for the multi-dimensional variable-order time fractional sub-diffusion equations, which have second order accuracy in time, second order and fourth order accuracy in space, respectively. The obtained difference schemes are proved to be uniquely solvable. The convergence and stability of the schemes in the discrete H 1 -norm are analyzed by utilizing the energy method. Some numerical examples are presented to verify the theoretical results.

In this paper, we study the classical problem of the wind in the steady atmospheric Ekman layer with classical boundary conditions and the eddy viscosity is an arbitrary height-dependent function with a finite limit value. We present existence and uniqueness and smooth results to justify computing first order approximation of solutions. Using a different argument that in previous works, we construct the Green’s function to derive the solution by a perturbation approach.

This paper proposes a stochastic model for mixed traffic flow with human-driven vehicles (HVs) and automated vehicles (AVs). The model is formulated in Lagrangian coordinates considering the heterogeneous behavior of human drivers. We further derive a first and second order approximation of the stochastic model describing the mean and the covariance dynamics, respectively, under different combinations of HVs and AVs in the traffic stream (e.g., randomly distributed in the stream, at the front of the stream, in the middle of the stream and in the rear of the stream). The proposed model allows us to explicitly investigate the interaction between AVs and HVs considering the uncertainty of human driving behavior. Six performance metrics are proposed to measure the impact of AVs on the uncertainty of HVs’ behavior, as well as on the stability of the system. The numerical experiment results show that AVs have significant impact on the uncertainty and stability of the mixed traffic flow system. Larger AV penetration rates can reduce the uncertainty inherent in HV behavior and improve the stability of the mixed flow substantially. Whereas AVs’ reaction time only has subtle impact on the uncertainty of the mixed stream; as well as the position of AVs in the traffic stream has marginal influence in terms of reducing uncertainty and improving stability.

In this paper, we propose a framework of Inexact Proximal Stochastic Second-order (IPSS) method for solving nonconvex optimization problems, whose objective function consists of an average of finit...

International audience

In this paper, we will deal with the problem of calculating the average velocity of a celestial object revolving around another celestial object in an elliptical orbit. After proving our main theorem to this effect, we will give some alternate forms of the formula for the average velocity, and show that this average value is in fact, attained at certain points of the orbit. We will conclude the paper by providing an intuitively natural and straightforward amendment of this formula.

Summary Low- and moderate-income (LMI) households remain less likely to adopt rooftop solar photovoltaics (PV) than higher-income households. A transient period of inequitable adoption is common among emerging technologies but stakeholders are calling for an accelerated transition to equitable rooftop PV adoption. To date, researchers have focused on demand-side drivers of PV adoption inequity, but supply-side factors could also play a role. Here, we use quote data to explore whether PV installers implement income-targeted marketing and the extent to which such strategies drive adoption inequity. We find that installers submit fewer quotes to households in low-income areas and those households that receive fewer quotes are less likely to adopt. The data suggest that income-targeted marketing explains about one-quarter of the difference in PV adoption rates between LMI and higher-income households. Policymakers could explore a broader suite of interventions to address demand- and supply-side drivers of PV adoption inequity., Graphical abstract, Highlights • Rooftop solar installers submit fewer quotes to low-income households • Income-targeted marketing could pose barriers to equitable solar adoption • Customers that receive fewer quotes are less likely to adopt solar • Policymakers could design low-income solar programs to address supply-side barriers, Energy resources; Energy policy; Energy Systems

This paper considers the out-of-sample forecasting performance of first- and second-order perturbation approximations for DSGE models that incorporate Markov-switching behaviour in the policy reaction function and the volatility of shocks. The results suggest that second-order approximations provide an improved forecasting performance in models that do not allow for regime-switching, while for the MS-DSGE models, a first-order approximation would appear to provide better out-of-sample properties. In addition, we find that over short-horizons, the MS-DSGE models provide superior forecasting results when compared to those models that do not allow for regime-switching (at both perturbation orders).

We develop a New Keynesian model featuring staggered price and wage contracts to study welfare costs of exogenous variations in trend inflation. The analyses show that the consequences of constant positive trend inflation and shifting trend inflation are severe, especially when trend inflation is high. Among two channels, staggered wage contracts play a vital role in transmitting adverse impacts of constant and shifting trend inflation into the economy. Without the staggered wage channel, these costs are modest. We also conduct exercises to examine the sensitivity of welfare costs to a wide range of plausible parameters. The results show that if the price and wage friction are sufficiently large, the price and wage indexation level are sufficiently small, or there is upward biased trend inflation process, the welfare costs become larger.

This volume is an on-line reprint of the original book published 1972 by Springer-Verlag which was intended to provide a comprehensive treatment of contemporary developments in methods of perturbation for nonlinear systems of ordinary differential equations. In this respect, it appeared to be a unique work, with hundreds of citations. Even today is a basic reference in the approximate solution of non-linear differential equations, specially appearing in problems of Celestial Mechanics. The original goal was to describe perturbation techniques, discuss their advantages and limitations and give some examples. The approach was founded on analytical and numerical methods of nonlinear mechanics. Attention had been given to the extension of methods to high orders of approximation, required now by the increased accuracy of measurements in all fields of science and technology. The main theorems relevant to each perturbation technique were outlined, but they only provided a foundation and were not the objective of the original book. Each chapter concluded with a detailed survey of the contemporary literature, supplemental information and more examples to complement the text, when necessary, for better comprehension.

In our paper (Prohens and Torregrosa, 2013 [1] ) we are concerned with the problem of shape and period of isolated periodic solutions of perturbed analytic radial Hamiltonian vector fields in the plane. Actually, there is a mistake in the formula of the first order approximation of the period given in Corollary 4 . Here we give its proper drafting.

The Young/Daly formula for periodic checkpointing is known to hold fora divisible load application where one can checkpoint at any time-step. In an nutshell, the optimal period isPYD=√2μfC where μf is the Mean Time Between Failures (MTBF) and C is the checkpoint time. This paper assesses the accuracy of the formulafor applications decomposed into computational iterations where: (i) the duration of aniteration is stochastic, i.e., obeys a probability distribution law D of mean μD ; and (ii) onecan checkpoint only at the end of an iteration. We first consider static strategies where checkpoints are taken after a given number of iterationskand provide a closed-form,asymptotically optimal, formula fork, valid for any distribution D. We then show that using the Young/Daly formula to compute k (ask·μD=PYD) is a first order approximationof this formula. We also consider dynamic strategies where one decides to checkpoint at the end of an iteration only if the total amount of work since the last checkpoint exceeds a threshold Wth, and otherwise proceed to the next iteration. Similarly, we provide aclosed-form formula for this threshold and show that PYD is a first-order approximation of Wth. Finally, we provide an extensive set of simulations where D is either Uniform,Gamma or truncated Normal, which shows the global accuracy of the Young/Daly formula,even when the distribution D had a large standard deviation (and when one cannot use afirst-order approximation). Hence we establish that the relevance of the formula goes well beyond its original framework; La formule de Young/Daly pour le checkpoint périodique s’applique aux applications divisibles où l’on peut prendre un checkpoint à tout instant. La période optimal est PYD=√2μfC où μf est le Mean Time Between Failures(MTBF) et C le temps de checkpoint. Ce rapport étudie la précision de cette formule pour des applications itératives telles que: (i) la durée d’une itération est stochastique, obéissant une distribution de probabilité D de moyenne μD; et (ii) on en peut prendre un checkpoint qu’à la fin d’une itération. Nous considérons des stratégies statiques (où on prend un checkpoint toutes les k itérations) et dynamiques (où on prend un checkpoint une fois un travail d’une certaine durée effectuée). Dans les deux cas, nous montrons que la formule de Young/Daly conduit au meilleur choix. Nous confirmons ces résultats théoriques par un large jeu de simulations pour des distributions Uniformes, Gamma ou Normale tronqué, montrant ainsi que la pertinence de la formule va bien au delà de sa formulation originelle.

We prove a discrete version of the first Weber inequality on three-dimensional hybrid spaces spanned by vectors of polynomials attached to the elements and faces of a polyhedral mesh. We then introduce two Hybrid High-Order methods for the approximation of the magnetostatic model, in both its (first-order) field and (second-order) vector potential formulations. These methods are applicable on general polyhedral meshes with star-shaped elements, and allow for arbitrary orders of approximation. Leveraging the previously established discrete Weber inequality, we perform a comprehensive analysis of the two methods. We finally validate them on a set of test-cases.

International audience; We introduce a three-dimensional Hybrid High-Order method for magnetostatic problems. The proposed method is easy to implement, supports general polyhedral meshes, and allows for arbitrary orders of approximation.

Time fractional order partial derivative equations are the generalization of the traditional integer order partial derivative equations, in which the time fractional order derivative is used to replace the corresponding integer order derivative. In this paper, the variable order time fractional diffusion-wave equation is considered. Firstly, numerical approximation scheme has been proposed by using the method of the piecewise linear interpolation join with a second order approximation of the first order time derivative to discrete the Coimbra variable order time fractional derivative. Then, central difference is used to approximate the second order of the spatial fractional derivatives. Finally, a numerical example has been presented to verify the numerical technique, which shows the efficiency of the numerical method.

Integrability condition of Hamiltonian perturbations of integrable Hamiltonian PDEs of hydrodynamic type up to the second order approximation is considered., 17 pages

In this paper, we present two new methods for solving convex mixed-integer nonlinear programming problems based on the outer approximation method. The first method is inspired by the level method and uses a regularization technique to reduce the step size when choosing new integer combinations. The second method combines ideas from both the level method and the sequential quadratic programming technique and uses a second order approximation of the Lagrangean when choosing the new integer combinations. The main idea behind the methods is to choose the integer combination more carefully at each iteration, in order to obtain the optimal solution in fewer iterations compared to the original outer approximation method. We prove rigorously that both methods will find and verify the optimal solution in a finite number of iterations. Furthermore, we present a numerical comparison of the methods based on 109 test problems to illustrate their advantages.

In many papers the first order approximation of the theoretical mean square error has been used for the ratio type estimators for the population mean and variance. The main focus of this paper is on examining the adequacy of the first order approximation and also on examining the robustness of estimators. We have calculated the theoretical mean square errors for many ratio type estimators, based on the first order approximation, and the corresponding empirical mean square errors. We observe that the first order approximation for ratio type mean estimators works well if the sampling fraction is small. We also observe that departure from the assumption of bivariate normality is not a serious handicap for large samples.

espanolSe propone la solucion numerica de la ecuacion de difusion no estatica unidimensional, desarrollando un algoritmo en software Matlab version 7.0, para lo cual se combina el esquema de diferencias finitas mimeticas en la aproximacion de los operadores diferenciales del continuo (gradiente y divergencia) para la variable espacial, sobre una malla uniforme, cuyos operadores diferenciales discretos presentan una aproximacion de segundo orden y el enfoque en diferencias finitas tipo Crank-Nicolson para obtener aproximaciones en la variable temporal. Este algoritmo propuesto para los enfoques mimeticos y Crank-Nicolson presentan mejor aproximacion que el esquema en diferencias finitas tipo Crank- Nicolson. Ademas se calcula el error de aproximacion generado entre la solucion numerica y la solucion analitica usando la norma del maximo para la ecuacion de difusion no estacionaria con condiciones de frontera tipo Robin. EnglishThe numerical solution of the one-dimensional non-static diffusion equation is proposed, developing an algorithm in software Matlab version 7.0, for which the mimetic finite difference scheme is combined in the approximation of the differential operators of the continuum (gradient and divergence) for the spatialvariable, on a uniform grid, whose discrete differential operators have a second order approximation and the finite difference approach type Crank Nicolson for approximations in the temporary variable.This proposed algorithm for the mimetic and Crank Nicolson approaches has a better approximation than the finite-difference Crank Nicolson-type scheme.In addition, the approximation error generated between the numerical solution and the analytical solution is calculated using the maximum standard for the non-stationary diffusion equation with Robin type boundary conditions.

In this paper, we consider the problem of sequentially estimating the unknown parameter in a particular exponential family of distributions under an asymmetric LINEX loss function and fixed cost for each observation within a Bayesian framework. Under a gamma prior distribution, the second order approximation for the Bayes risks of the asymptotically pointwise optimal rule and the optimal stopping rule are derived. It is shown that the asymptotically pointwise optimal rule is asymptotically non-deficient in the sense of Woodroofe (1981).

By using the perturbation expansion method and self-consistent iterative method, we evaluate the effect of the conduction band nonprabolicity on the wave vector ( k t ) dependent Rashba coefficient ( α ) and nonlinear Rashba spin splitting (Δ E ) in the Al 0.5 Ga 0.5 N/GaN quantum well (QW). The effective mass (energy) under the first order approximation m t1 ( E k1 ) is in proximity to the iterative result m tp ( E kp ) and m t1 > m tp , E k1 E kp , showing the higher order contributions to m t ( E k ) are small. The sign of the nonparabolic correction to E k is just opposite to that of the correction to m t . The increase of α and Δ E due to the conduction band nonparabolicity reaches about 3% at k t =1 nm −1 . Around the left heterointerface, the probability density is high and E k0 > E kp > E k1 , so α 0 α p α 1 , Δ E 0 E p E 1 . With increasing k t , α decreases, and Δ E increases slowly. For small k t , α 0 (Δ E 0 ), α 1 (Δ E 1 ) and α p (Δ E p ) are nearly the same. While for large k t , the difference between α 0 and α 1 ( α p ) increases rapidly, but the difference between Δ E 0 and Δ E 1 (Δ E p ) increases slowly.

This paper aims to summarizing the different approaches in determining the implied volatility for the options. This value is of particular importance since it is the main component of theoptions price and because, among traders, options are quoted in terms of volatility rather than price. After a discussion on the approximation methods, a numerical approach is explained. It is shown that, in order to ensure a fast and reliable convergence, the selection of an appropriate starting point is key. The authors suggestion for choosing the first order approximation or the inflexion as initial point is also illustrated.

In real life customers are motivated to buy more if there is more availability of product. In this study, we develop inventory model for inventory dependent demand with different holding cost function. Mathematical formulations are discussed for two situations i.e., power demand rate with inventory dependent holding cost functions and inventory-induced demand rate with time and inventory dependent holding cost functions. The second order approximation is applied for exponential terms to find numerical values. Numerical examples are discussed to illustrate the models proposed in this study. The sensitivity analysis is given for optimal (minimum) solution with respect to key parameters is also discussed.

We use Monte Carlo experiments to evaluate whether “upward pricing pressure” (UPP) accurately predicts the price effects of mergers, motivated by the observation that UPP is a restricted form of the first order approximation derived in Jaffe and Weyl (2013). Results indicate that UPP is quite accurate with standard log-concave demand systems, but understates price effects if demand exhibits greater convexity. Prediction error does not systematically exceed that of misspecified simulation models, nor is it much greater than that of correctly-specified models simulated with imprecise demand elasticities. The results also support that UPP provides accurate screens for anticompetitive mergers.

A radiotherapy treatment margin formula has been analytically derived when a standard deviation (SD) of systematic positioning errors Ʃ is relatively small compared to an SD of random positioning errors σ. The margin formula for 0 ≤ Ʃ ≤ σ was calculated by linearly interpolating two boundaries at Ʃ = 0 and Ʃ = σ, assuming that the van Herk margin approximation of k1Ʃ + k2σ is valid at Ʃ = σ. It was shown that a margin formula for 0 ≤ Ʃ ≤ σ may be approximated by k1σ + k2Ʃ, leading to a more general form of k1 max(Ʃ,σ) + k2 min(Ʃ,σ) which is a piecewise linear approximation for any values of Ʃ and σ.

In the present work, the triple and double cross sections of atomic hydrogen ionization in collision with the proton are calculated. The potential of the interaction is considered as a Coulombic; also, the first and second order Born approximations have been used. The singularity of the Green operator in the second order approximation has been removed for the analytical solution of the scattering amplitude and the triple cross section calculation. Finally, the results for triple and double cross sections have been compared with the available theoretical and experimental results. The first and the second Born approximations are usually applied at intermediate and high impact energy ranges. However, it is shown that it could be successfully applied at lower energies as well.

This paper proposed a new integral inequality, called second-order approximation integral inequality (SAII), that could significantly reduce the conservativeness in stability analysis of systems with time delays. The former well-known integral inequalities such as Jensen's inequality and Wirtinger based inequality, are all included in the proposed integral inequality as special cases. Furthermore, it's shown that Jensen's and Wirtinger based inequalities are just zero-order and first-order approximation, respectively. Stability criterion with less conservatism is then developed using SAII for time delay systems. Numerical examples are given to demonstrate the effectiveness and benefit of the proposed method.

Previous studies of electronic transport in molybdenum disulfide (MoS2) are restricted to the first order approximation of the Hamiltonian. In this paper, to obtain more exact results, we firstly present an analytical solution for the Hamiltonian of MoS2 when terms up to the second order (quadratic k-dependent) of the Hamiltonian are taken into account. Our analytical solution is easily applicable to study the transport properties of any single and multi-junctions of MoS2. Then, we propose a device composed of two ferromagnetic barriers with anti-parallel exchange fields and we show that this device has interesting properties such as spin and valley filtering with perfect spin and valley polarizations. Using this device, we can easily switch both spin and valley polarizations to their opposite polarizations only by using electric voltage, which is of fundamental importance in quantum computation and the next generation of logic devices.

In this paper, we introduce two new particle filtering algorithms for high-dimensional state spaces in the multiple particle filtering approach. In multiple particle filtering, the state space is partitioned and a different particle filter is used for each component of the partition. At each time step, all particle filters share information about their marginal densities so that they can adequately approximate the filtering recursion. In this paper, we propose a second order approximation to the involved densities based on sigma-point integration methods. We then introduce two different particle filters that make use of this strategy. Finally, we demonstrate their remarkable performance through simulations of a multiple target tracking scenario with a sensor network.

In this paper, we consider the problem of missing complete at random (MCAR) values in two-phase probability proportional to size ( pps) sampling for the estimation of population mean. For comprehensive examination of missing values, we discuss four possible situations for the occurrence of non-response. A class of estimators is considered by using an unknown constant with the traditional estimators for imputing the missing values. Theoretically, bias and mean squared errors of the proposed estimators are obtained up to the first order approximation. Two numerical studies are carried out for the relative comparison of the proposed estimators with mean estimator under two-phase ppssampling for each of the predefined situation.

Breaking the uncovered interest rate parity (UIP) condition is essential to accounting for the empirical behavior of exchange rates, and is a prerequisite for theoretical analysis of sterilized foreign exchange interventions. Gabaix and Maggiori (2015) account for some of the long-standing empirical exchange rate puzzles by introducing financial intermediaries that are willing to absorb international saving imbalances for a premium, thereby deviating from the UIP. In another important contribution, Fanelli and Straub (2019) lay down the principles for foreign exchange interventions. In their model, regulatory exposure limits and participation cost in the international financial markets drive a wedge in the UIP. This paper demonstrates that, to a first order approximation, these models are equivalent to a reduced-form portfolio adjustment cost model, as in Schmitt-Grohe and Uribe (2003). Therefore, to the extent that one is only concerned with first-order dynamics and second moments, there is no gain from adopting the rich microstructure of either models -- a simple portfolio adjustment cost is just as good.

This paper contributes by providing a new approach to study optimal macroprudential policies based on economy wide welfare. Following Gerba (2017), we pin down a welfare function based on a first-and second order approximation of the aggregate utility in the economy and use it to determine the merits of different macroprudential rules for Euro Area. With the aim to test this framework, we apply it to the model of Clerc et al. (2015). We find that the optimal level of capital is 15.6 percent, or 2.4 percentage points higher tan the 2001-2015 value. Optimal capital reduces significantly the volatility of the economy while increasing somewhat the total level of welfare in steady state, even with a time-invariant instrument. Expressed differently, bank default rates would have been 3.5 percentage points lower while credit and GDP 5% and 0.8% higher had optimal capital level been in place during the 2011-2013 crisis. Further, using a model-consistent loss function, we find that the optimal Countercyclical Capital Buffer (CCyB) rule depends on whether observed or optimal capital levels are already in place. Conditional on optimal capital level, optimal CCyB rule should respond to movements in total credit and mortgage lending spreads. Gains in welfare from optimal combination of instruments is higher than the sum of their individual effects due to synergies and positive mutual spillovers.

For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn. We studied conformal transformations when the original space is a space of non-zero constant curvature.For the Riemannian space Vn, the invariantly associated with Vn space V˜n2 is constructed, which implements a second order approximation for Vn. We studied conformal transformations when the original space is a space of non-zero constant curvature.

In this paper, the vibration of a micro-electromechanical resonator with positive position feedback controller is studied. The analytical results are obtained to the first order approximation by using the multiple scale perturbation technique. The stability of the steady-state solution is presented and studied applying frequency response equations near the simultaneous primary and internal resonance cases. The effects of the controller and some system parameters on the vibrating system are studied numerically. The main result of this paper indicates that it is possible to reduce the vibration for the resonator system.

We use Monte Carlo experiments to study how pass‐through can improve merger price predictions, focusing on the first order approximation (FOA) proposed in Jaffe and Weyl [[Jaffe, S., 2013]]. FOA addresses the functional form misspecification that can exist in standard merger simulations. We find that the predictions of FOA are tightly distributed around the true price effects if pass‐through is precise, but that measurement error in pass‐through diminishes accuracy. As a comparison to FOA, we also study a methodology that uses pass‐through to select among functional forms for use in simulation. This alternative also increases accuracy relative to standard merger simulation and proves more robust to measurement error.

A second order scheme is constructed for the scalar conservation laws with flux function allowed to be discontinuous in the space variable. The corresponding numerical solutionis shown to converge to the (A,B) entropysolution. Numerical results are presented to validate the convergence analysis. In addition, it is inferred that these numerical solutions are comparable with those obtained using the standard minmod limiter.

Within the optimal production and hedging decision framework, Lien compares the exponential utility function with its second order approximation under the normality distribution assumption. In this paper, we first extend the result further by comparing the exponential utility function with a 2n-order approximation for any integer n. We then propose an approach with illustration to find the smallest n that provides a good approximation.