In , Zhang considered a linear conservativescheme for GRLW equation, however, the accuracy of the scheme is only second-order. Recently, there has been growing in- terest in high-order compact methods to solve the partial diﬀerential equations [– ], where fourth-order compact ﬁnite diﬀerence approximation solutions for the tran- sient wave equations, a N-carrier system, the Klein-Gordon equation, the Sine-Gordon equation, the one-dimensional heat and advection-diﬀusion equations, the Schrödinger equation, the Klein-Gordon-Schrödinger equation and the RLW equation were shown, respectively. These numerical methods may give us many enlightenments to design a new numerical scheme for the GRLW equation. For a wide and most complete vision concern- ing the importance, the breadth, and the interest of the topics covered, we should also recall the study done on the long waves in [–].
This equation is usually called the Rosenau-RLW equation. The Rosenau-RLW equation (.) has been solved numerically by various methods. Zuo et al.  have proposed a non- linear implicit conservativescheme for the general Rosenau-RLW equation. In [, ], Pan et al. have presented three-level linearized diﬀerence schemes for both (.) and the gen- eral Rosenau-RLW equation. Atouani and Omrani have developed a Galerkin ﬁnite ele- ment method for (.) . Hu and Wang have also proposed a high-accuracy linear conser- vative scheme to solve (.) . When γ = β = , system (.) is reduced to the Rosenau equation :
If a data packet is lost during transmission at one link, such packet has to be re transmitted, which will consume some extra energy. Therefore, this mode includes the energy consumption for both the new data packet and the retransmitted packet. The authors in  proposed a minimum total reliable transmission power routing protocol. All the existing cost models, however, ignore additional energy consumption in exchanging control (or signaling) packets at the Data Link layer, which therefore underestimates the actual energy consumption with various wireless protocols. Unlike cellular networks, the lifetime of mobile nodes will deeply impact on the performance of ad hoc networks. In a cellular network, a reduction in the number of active mobile nodes will reduce the amount of signal interference and channel contentions. However, since the mobile nodes in an ad hoc network need to relay their packets through the other mobile nodes toward the intended destinations, a decrease in the number of participating mobile nodes may lead to the network disconnected, there by hurting the performance of the network. In this paper, we analyze the energy consumption to achieve reliable transmission, and propose more accurate scheme for energy conservation along with security measures. We verify the accuracy of the proposed scheme and also demonstrate the usefulness of the accurate model in achieving more energy efficient routing in 802.11 based networks
Another high-resolution scheme called the CIP-MOCCT scheme (Kudoh and Shibata, 1997) for solving the MHD equations, which is based on the Constrained Interpolation Proﬁle (CIP) scheme (Yabe et al., 2001) and the Method Of Characteristics-Constrained Transport (MOCCT) scheme (Hawley and Stone, 1995). The CIP scheme has been de- veloped for solving hyperbolic equations. The MOCCT scheme solves the induction equation and magnetic stress in the MHD equation, maintaining the divergence-free condi- tion of the magnetic ﬁeld i.e., ∇ · B = 0. However, there are several problems associated with the CIP-MOCCT scheme. It needs to add a certain diffusion term to suppress numeri- cal oscillation, and it also needs a larger computer memory than other high-resolution schemes for the MHD equations in order to store partial derivatives or integrals of a proﬁle.
schemes (the solid lines) and the non-conservative schemes (the dashed lines). The parameters in the computations are the same as described in the previous Sect. 4.2. To make fair comparison, all other aspects are kept same, including the time-step scheme (TR-BDF2), the discretization scheme for interior points of the domain, the initial condition, the phys- ical parameters, the time-step size t and the space resolu- tion x . As shown in Fig. 3(a), the ion concentration from the non-conservativescheme is substantially lower than that from the mass-conservativescheme and the variations near the boundaries are much smaller in the result from the non- conservativescheme. Furthermore, the electrostatic poten- tial obtained from the non-conservativescheme, shown in Fig. 3(b), has a linear profile with non-zero slope in the middle of the domain and much milder slopes at the bound-
It is known the conservativescheme is better than the nonconservative ones. Zhang et al. 1 point out that the nonconservative scheme may easily show nonlinear blow up. In 2 Li and Vu-Quoc said “. . . in some areas, the ability to preserve some invariant properties of the original diﬀerential equation is a criterion to judge the success of a numerical simulation”. In 3–11 , some conservative finite diﬀerence schemes were used for a system of the generalized nonlinear Schr ¨odinger equations, Regularized long wave RLW equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative diﬀerence scheme for the general Rosenau-RLW equation, which simulates conservative laws 1.4 and 1.5 at the same time. The outline of the paper is as follows. In Section 2, a nonlinear diﬀerence scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
We will illustrate the method by describing a mass-conservativescheme (i.e. preserving ion concentration exactly) for solving the nonlinear systems of PDEs (13) and (14). The ex- tension of the method to the multi-dimensional case is straightforward. This scheme uses the trapezoidal rule and the second-order backward differentiation formula (TR-BDF2) in time and the second-order central differencing in space. The TR-BDF2 scheme is implicit in time, resulting in a system of nonlinear equations after discretization. Instead of using the Newton-Raphson method for solving the large nonlinear systems at each time step, we present a simple iterative scheme which is easy to implement and can solve the systems efficiently.
line) and the non-conservative schemes (the dotted line) obtained by using a second-order finite differ- ence based on the numerical result E(t) shown in Figure 5(a). In the same graph, we also plot the expected dissipation rate given by the right-hand side of (10), computed using the second-order cen- tral differencing and trapezoidal rule and shown by the dashed line for the conservativescheme and the dash-dotted line for the non-conservativescheme in Fig. 5(b). It shows that the numerical result from the conservativescheme (the solid line) agrees with the energy dissipation law (the dashed line) very well. In contrast, the corresponding results for the non-conservativescheme show that the energy dis- sipation law is not satisfied after a short period of time. This is due to the fact that the total concentra- tion from the non-conservativescheme displays very poor performance in conserving the total concentra- tions. The results show that the discretization of the boundary conditions have profound impact on satis- fying the physical properties: the energy dissipation law and the conservation of the total number of ions. 3
The conservativescheme needs the video size in the be- ginning to partition it into equal-sized segments and subsegments. Generally, in a live video we do not know the video size; so this scheme cannot be applied in its existing form. We modify its basic architecture. We do not divide the segments or video channels any further. For the modified conservativescheme, we discuss a me- chanism so that this scheme can support live video broadcasting. We assume that the bandwidth allocated to the video is finite. This assumption is not illogical be- cause for abundant bandwidth there is hardly any issue to discuss.
with the reference solution very well and there is no obvi- ous difference between these two contour plots. Compared with the results of our former fourth-order model, the con- tour lines look slightly less smooth. Similar results are found in the spectral transform reference solution. Since this test contains more significant gradients in the solution, a high- order scheme might need some extra numerical dissipation to remove the noise around the large gradients. Increasing the grid solution can effectively reduce the magnitude of the oscillations as shown in the present simulation.
the maximum norm in . In , a new conservative diﬀerence scheme for the general Rosenau-RLW equation was proposed. In , Pan and Zhang proposed a conservative linearized diﬀerence scheme for the general Rosenau-RLW equation which was uncondi- tionally stable and second-order convergent and simulates conservative laws at the same time. In , the initial-boundary value problem for the Rosenau-RLW equation was stud- ied. One proposed a three-level linear ﬁnite diﬀerence scheme, which has the theoretical accuracy of O(τ + h ).
The Czech and Slovak models, by contrast, are much more traditional and stable with the predominant role of the incumbent ČD and ŽSR respectively, which are still in the hands of the state and hold positions of national operators. The Slovak model can be described as particularly conservative. Apart from some open access Czech companies which are active on the long-distance lines to Prague, only one external carrier is present in Slovakia. Moreover, the organization and financing of regional railway services remains the responsibility of the national government. While the scale of market opening in the Czech Republic is larger than in Slovakia, if compared with some Western-European countries it can be described as very small still. The role of the incumbent is absolutely predominant and it does not seem that this is going to change any time soon as all Czech regions have decided to sign long- term direct contracts with ČD.
Abstract. Lagrangian particle dispersion models require in- terpolation of all meteorological input variables to the posi- tion in space and time of computational particles. The widely used model FLEXPART uses linear interpolation for this pur- pose, implying that the discrete input fields contain point val- ues. As this is not the case for precipitation (and other fluxes) which represent cell averages or integrals, a preprocessing scheme is applied which ensures the conservation of the in- tegral quantity with the linear interpolation in FLEXPART, at least for the temporal dimension. However, this mass con- servation is not ensured per grid cell, and the scheme thus has undesirable properties such as temporal smoothing of the precipitation rates. Therefore, a new reconstruction algo- rithm was developed, in two variants. It introduces additional supporting grid points in each time interval and is to be used with a piecewise linear interpolation to reconstruct the pre- cipitation time series in FLEXPART. It fulfils the desired re- quirements by preserving the integral precipitation in each time interval, guaranteeing continuity at interval boundaries, and maintaining non-negativity. The function values of the reconstruction algorithm at the sub-grid and boundary grid points constitute the degrees of freedom, which can be pre- scribed in various ways. With the requirements mentioned it was possible to derive a suitable piecewise linear reconstruc- tion. To improve the monotonicity behaviour, two versions of a filter were also developed that form a part of the final al- gorithm. Currently, the algorithm is meant primarily for the temporal dimension. It was shown to significantly improve the reconstruction of hourly precipitation time series from 3-hourly input data. Preliminary considerations for the ex-
It is an important basis for validating the numerical method whether the scheme can capture the dam-break bore waves accurately or not. This g ives rise to an increasing interest in solving such a proble m. Fro m 1980 to 2000 several fin ite-difference schemes that handle discontinuities effectively we re used to compute open -channel flo ws, such as the approximate Rie mann solver [3,4]. Based on the above research results, the goal of the current work is to develop a mathe matica l mode l capable of dea ling with hydraulic discontinuities such as steep fronts, hydraulic ju mp and drop, etc. The water governing equations has been solved by the WENO scheme and the Finite Vo lu me Method on unstructured grid.
This paper is devoted to fill in the lack of result for conservative finite difference scheme for problem (1) and nonsymmetric sparse block matrices related to finite difference equations in polar coordinates. We present a conservative finite difference scheme for this problem and prove its convergence. Our approach is based on the Lax-Wondroff theorem , which guarantees convergence of a conservative FD schemes in the class of weak solutions, as the polar mesh is refined. Note that a similar technique was used in  for problem (1), where the classical solution of the boundary value problem (1) is considered. Since we are interested in bounded weak solutions u ∈ H 1 (Ω R ), we require that the solution of problem (1) satisfies the boundedness at r = 0 condition
2013: 345). Cameron also had to contend with widespread Euroscepticism in the second arena: the party organization. Here, a 2013 survey of party members found some 70.8 percent favouring withdrawal from the EU, although 53.6 percent were willing to back remaining after a renegotiation of the terms of membership (Bale and Webb 2016: 126). Eurosceptic sentiment could also be identified as a threat to the Conservatives in the third arena of the mass electorate. Under the leadership of Nigel Farage the UK Independence Party (UKIP) made significant advances in the opinion polls, particularly following the unpopular March 2012 budget (which was labelled an ÔomnishamblesÕ). The fact that the Conservatives were in Coalition with the Liberal Democrats created political space to their right which UKIP were keen to exploit, and fuelled pressure within the Conservative Party for Cameron to try and counter their appeal (Lynch and Whitaker 2016: 128). This (historically unusual) competition on the right of British politics was illustrated by survey data suggesting that more than half of Conservative Party members Ð who it could
The flux across a cell interface is based on the integral solution of the model equa- tion. Discontinuous spatial reconstruction with nonlinear limiter is used to introduce artificial dissipation for UGKS once the scheme becomes a shock capturing method when the dissipative flow structure cannot be well resolved by the cell size. Details can be found in . In this paper, we use van Leer limiter in the reconstruction. Due to the discreteness of the velocity space, numerical quadrature should be used to calculate various integrals. In this paper, composite Newton-Cote ’ s (N − C) quadrature is adopted.