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## Nonlinear Terms (nonlinear + term)
## Selected Abstracts## Numerical implementation of the Crank,Nicolson/Adams,Bashforth scheme for the time-dependent Navier,Stokes equations INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 6 2010Yinnian HeAbstract This article considers numerical implementation of the Crank,Nicolson/Adams,Bashforth scheme for the two-dimensional non-stationary Navier,Stokes equations. A finite element method is applied for the spatial approximation of the velocity and pressure. The time discretization is based on the Crank,Nicolson scheme for the linear term and the explicit Adams,Bashforth scheme for the nonlinear term. Comparison with other methods, through a series of numerical experiments, shows that this method is almost unconditionally stable and convergent, i.e. stable and convergent when the time step is smaller than a given constant. Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Numerical simulation of the miscible displacement of radionuclides in a heterogeneous porous medium INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 10 2005C.-H. BruneauAbstract The aim of this paper is to model and simulate the displacement of radioactive elements in a saturated heterogeneous porous medium. New schemes are proposed to solve accurately the convection,diffusion,reaction equations including nonlinear terms in the time derivative. Numerical tests show the stability and robustness of these schemes through strong heterogeneities of the medium. Finally the COUPLEX 1 benchmark concerning the far field simulation of a polluted flow by a leak of a nuclear waste disposal is performed and compared with the results available in the literature. Copyright © 2005 John Wiley & Sons, Ltd. [source] ## Observer design for nonlinear discrete-time systems: Immersion and dynamic observer error linearization techniques INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 5 2010Jian ZhangAbstract This paper focuses on the observer design for nonlinear discrete-time systems by means of nonlinear observer canonical form. At first, sufficient and necessary conditions are obtained for a class of autonomous nonlinear discrete-time systems to be immersible into higher dimensional observer canonical form. Then a method called dynamic observer error linearization is developed. By introducing a dynamic auxiliary system, the augmented system is shown to be locally equivalent to the generalized observer form, whose nonlinear terms contain auxiliary states and output of the system. A constructive algorithm is also provided to obtain the state coordinate transformation. These results are an extension of their counterparts of nonlinear continuous-time systems to nonlinear discrete-time systems (Syst. Control Lett. 1986; 7:133,142; SIAM. J. Control Optim. 2003; 41:1756,1778; Int. J. Control 2004; 77:723,734; Automatica 2006; 42:321,328; IEEE Trans. Automat. Control 2007; 52:83,88; IEEE Trans. Automat. Control 2004; 49:1746,1750; Automatica 2006; 42:2195,2200; IEEE Trans. Automat. Control 1996; 41:598,603; Syst. Control Lett. 1997; 31:115,128). Copyright © 2009 John Wiley & Sons, Ltd. [source] ## Stabilization of complex cascade systems using boundedness information in finite time INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, Issue 10 2009Huawen YeAbstract In this paper, the stabilization problem of several classes of complex cascade systems is investigated from a new point of view. If the closed-loop system is proven to have no finite escape time, the boundedness information in finite time, which is obtained from robust stable subsystems or recursive analysis procedures, is then sufficiently employed to deal with crucial nonlinear terms. The proposed method does not rely on complicated Lyapunov functions, and in some cases it can avoid strong growth conditions and complicated small gain analysis. In addition, simple saturated control laws are explicitly constructed in an almost unified way. Copyright © 2008 John Wiley & Sons, Ltd. [source] ## The transient equations of viscous quantum hydrodynamics MATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 4 2008Michael DreherAbstract We study the viscous model of quantum hydrodynamics in a bounded domain of space dimension 1, 2, or 3, and in the full one-dimensional space. This model is a mixed-order partial differential system with nonlocal and nonlinear terms for the particle density, current density, and electric potential. By a viscous regularization approach, we show existence and uniqueness of local in time solutions. We propose a reformulation as an equation of Schrödinger type, and we prove the inviscid limit. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Energy properties preserving schemes for Burgers' equation, NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 1 2008R. AnguelovAbstract The Burgers' equation, a simplification of the Navier,Stokes equations, is one of the fundamental model equations in gas dynamics, hydrodynamics, and acoustics that illustrates the coupling between convection/advection and diffusion. The kinetic energy enjoys boundedness and monotone decreasing properties that are useful in the study of the asymptotic behavior of the solution. We construct a family of non-standard finite difference schemes, which replicate the energy equality and the properties of the kinetic energy. Our approach is based on Mickens' rule [Nonstandard Finite Difference Models of Differential Equations, World Scientific, Singapore, 1994.] of nonlocal approximation of nonlinear terms. More precisely, we propose a systematic nonlocal way of generating approximations that ensure that the trilinear form is identically zero for repeated arguments. We provide numerical experiments that support the theory and demonstrate the power of the non-standard schemes over the classical ones. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007 [source] ## Method of lines with boundary elements for 1-D transient diffusion-reaction problems NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2006P.A. RamachandranAbstract Time-dependent differential equations can be solved using the concept of method of lines (MOL) together with the boundary element (BE) representation for the spatial linear part of the equation. The BE method alleviates the need for spatial discretization and casts the problem in an integral format. Hence errors associated with the numerical approximation of the spatial derivatives are totally eliminated. An element level local cubic approximation is used for the variable at each time step to facilitate the time marching and the nonlinear terms are represented in a semi-implicit manner by a local linearization at each time step. The accuracy of the method has been illustrated on a number of test problems of engineering significance. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2006 [source] ## Uniform stability of spectral nonlinear Galerkin methods NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 5 2004Yinnian HeAbstract This article provides a stability analysis for the backward Euler schemes of time discretization applied to the spatially discrete spectral standard and nonlinear Galerkin approximations of the nonstationary Navier-Stokes equations with some appropriate assumption of the data (,, u0, f). If the backward Euler scheme with the semi-implicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraint ,t , (2/,,1). Moreover, if the backward Euler scheme with the explicit nonlinear terms is used, the spectral standard and nonlinear Galerkin methods are uniform stable under the time step constraints ,t = O(,) and ,t = O(,), respectively, where , , ,, which shows that the restriction on the time step of the spectral nonlinear Galerkin method is less than that of the spectral standard Galerkin method. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2004 [source] ## An analytical model for the rapid intensification of tropical cyclones THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 642 2009Chanh Q. KieuAbstract The nonlinearity and complexity of the primitive equations have been key obstacles to our understanding of tropical cyclones (TCs), particularly in relation to the dynamical processes leading to their rapid intensification. In this study, an axisymmetric model, in which all nonlinear terms in the horizontal momentum equations are retained, is used to examine analytically the effects of organized deep convection on TC rapid intensification. By prescribing a vertical profile of the vertical motion with exponential growth in the core region, a class of exact time-dependent solutions for the primary circulations of TCs are obtained. The analytical solutions are shown to capture well many observed dynamical structures in both the core and outer regions and the rapid growth of TCs in terms of maximum winds and central pressure drops. The analytical solutions reveal that (1) the rotational flows in the inner-core region grow double-exponentially, and the central pressure drops occur at rates much faster than the rotational growth; (2) the amplification rates of the primary circulations differ profoundly from those of the secondary circulations; (3) the rotational flows tend to grow from the bottom upwards with the fastest growth occurring at the lowest levels; and (4) the TC growth rates depend critically on the vertical structure of tangential flows, with a faster rate for a lower-level peak rotation. The nonlinear dynamics are shown to play an important role in the rapid growth of TCs. It is demonstrated that the analytical solutions can also be used to construct dynamically consistent vortices for the initialization of TC models. Limitations and possible improvements of the analytical model are also discussed. Copyright © 2009 Royal Meteorological Society [source] ## Limitations of a linear model for the hurricane boundary layer THE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 641 2009Stefanie VoglAbstract The linear model for the steady boundary layer of a rapidly rotating axisymmetric vortex is derived from a detailed scale analysis of the full equations of motion. The previously known analytic solution is re-appraised for vortices of hurricane scale and strength. The internal consistency of the linear approximation is investigated for such a vortex by calculating from the solution the magnitude of the nonlinear terms that are neglected in the approximation compared with the terms retained. It is shown that the nonlinear terms are not negligibly small in a large region of the vortex, a feature that is consistent with the scale analysis. We argue that the boundary-layer problem is well-posed only at outer radii where there is subsidence into the layer. At inner radii, where there is ascent, only the radial pressure gradient may be prescribed and not the wind components at the top of the boundary layer, but the linear problem cannot be solved in these circumstances. We examine the radius at which the vertical flow at the top of the boundary layer changes sign for different tangential wind profiles relevant to hurricanes and show that this is several hundred kilometres from the vortex centre. This feature represents a further limitation of the linear model applied to hurricanes. While the present analysis assumes axial symmetry, the same limitations presumably apply to non-axisymmetric extensions to the linear model. Copyright © 2009 Royal Meteorological Society [source] ## Stabilization for a class of complex interlaced systems using asymptotical gain, ASIAN JOURNAL OF CONTROL, Issue 2 2010Huawen YeAbstract This paper addresses the stabilization problem of a class of interlaced systems that are not in a strict-feedforward form and contain some severe nonlinear terms. Bounded control laws in a fractional form are explicitly constructed. The feature of stability analysis allows the closed-loop system, when it is proven to have no finite escape time, to employ the asymptotical gain, which is obtained from an input-to-state stable (ISS) subsystem, to calculate the severe nonlinear terms, and the related estimate in turn guides the assignment of small controls. Together with the use of the passivity theory and the ISS-based stability criterion, the proposed method requires no small gain analysis although the asymptotical gain is used, and differs from the state-dependent saturation scheme since the controls do not include saturation functions. As an application, a new stabilizing control law is presented for the well-known friction ball-and-beam system. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society [source] ## Reduced modeling and state observation of an activated sludge process BIOTECHNOLOGY PROGRESS, Issue 3 2009Isabelle QueinnecAbstract This article first proposes a reduction strategy of the activated sludge process model with alternated aeration. Initiated with the standard activated sludge model (ASM1), the reduction is based on some biochemical considerations followed by linear approximations of nonlinear terms. Two submodels are then obtained, one for the aerobic phase and one for the anoxic phase, using four state variables related to the organic substrate concentration, the ammonium and nitrate-nitrite nitrogen, and the oxygen concentration. Then, a two-step robust estimation strategy is used to estimate both the unmeasured state variables and the unknown inflow ammonium nitrogen concentration. Parameter uncertainty is considered in the dynamics and input matrices of the system. © 2009 American Institute of Chemical Engineers Biotechnol. Prog., 2009 [source] |