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Spatial Derivatives (spatial + derivative)
Selected AbstractsDevelopment of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] An interpolation-based local differential quadrature method to solve partial differential equations using irregularly distributed nodesINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 7 2008Hang Ma Abstract To circumvent the constraint in application of the conventional differential quadrature (DQ) method that the solution domain has to be a regular region, an interpolation-based local differential quadrature (LDQ) method is proposed in this paper. Instead of using regular nodes placed on mesh lines in the DQ method (DQM), irregularly distributed nodes are employed in the LDQ method. That is, any spatial derivative at a nodal point is approximated by a linear weighted sum of the functional values of irregularly distributed nodes in the local physical domain. The feature of the new approach lies in the fact that the weighting coefficients are determined by the quadrature rule over the irregularly distributed local supporting nodes with the aid of nodal interpolation techniques developed in the paper. Because of this distinctive feature, the LDQ method can be consistently applied to linear and nonlinear problems and is really a mesh-free method without the limitation in the solution domain of the conventional DQM. The effectiveness and efficiency of the method are validated by two simple numerical examples by solving boundary-value problems of a linear and a nonlinear partial differential equation. Copyright © 2007 John Wiley & Sons, Ltd. [source] Third-order methods for first-order hyperbolic partial differential equationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 1 2004T. A. Cheema Abstract In this paper numerical methods for solving first-order hyperbolic partial differential equations are developed. These methods are developed by approximating the first-order spatial derivative by third-order finite-difference approximations and a matrix exponential function by a third-order rational approximation having distinct real poles. Then parallel algorithms are developed and tested on a sequential computer for an advection equation with constant coefficient and a non-linear problem. Copyright © 2003 John Wiley & Sons, Ltd. [source] Non-local dispersive model for wave propagation in heterogeneous media: multi-dimensional caseINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 3 2002Jacob Fish Abstract Three non-dispersive models in multi-dimensions have been developed. The first model consists of a leading-order homogenized equation of motion subjected to the secularity constraints imposing uniform validity of asymptotic expansions. The second, non-local model, contains a fourth-order spatial derivative and thus requires C1 continuous finite element formulation. The third model, which is limited to the constant mass density and a macroscopically orthotropic heterogeneous medium, requires C0 continuity only and its finite element formulation is almost identical to the classical local approach with the exception of the mass matrix. The modified mass matrix consists of the classical mass matrix (lumped or consistent) perturbed with a stiffness matrix whose constitutive matrix depends on the unit cell solution. Numerical results are presented to validate the present formulations. Copyright © 2002 John Wiley & Sons, Ltd. [source] The generalized Dirichlet-to-Neumann map for certain nonlinear evolution PDEsCOMMUNICATIONS ON PURE & APPLIED MATHEMATICS, Issue 5 2005A. S. Fokas Let q(x,t) satisfy a nonlinear integrable evolution PDE whose highest spatial derivative is of order n. An initial boundary value problem on the half-line for such a PDE is at least linearly well-posed if one prescribes initial conditions, as well as N boundary conditions at x = 0, where for n even N equals n/2 and for n odd, depending on the sign of the highest derivative, N equals either n,1/2 or n+1/2. For example, for the nonlinear Schrödinger (NLS) and the sine-Gordon (sG), N = 1, while for the modified Korteweg-deVries (mKdV) N = 1 or N = 2 depending on the sign of the third derivative. Constructing the generalized Dirichlet-to-Neumann map means determining those boundary values at x = 0 that are not prescribed as boundary conditions in terms of the given initial and boundary conditions. A general methodology is presented that constructs this map in terms of the solution of a system of two nonlinear ODEs. This formulation implies that for the focusing NLS, for the sG, and for the two focusing versions of the mKdV, this map is global in time. It appears that this is the first time in the literature that such a characterization for nonlinear PDEs is explicitly described. It is also shown here that for particular choices of the boundary conditions the above map can be linearized. © 2005 Wiley Periodicals, Inc. [source] On accuracy of the finite-difference and finite-element schemes with respect to P -wave to S -wave speed ratioGEOPHYSICAL JOURNAL INTERNATIONAL, Issue 1 2010Peter Moczo SUMMARY Numerical modelling of seismic motion in sedimentary basins often has to account for P -wave to S -wave speed ratios as large as five and even larger, mainly in sediments below groundwater level. Therefore, we analyse seven schemes for their behaviour with a varying P -wave to S -wave speed ratio. Four finite-difference (FD) schemes include (1) displacement conventional-grid, (2) displacement-stress partly-staggered-grid, (3) displacement-stress staggered-grid and (4) velocity,stress staggered-grid schemes. Three displacement finite-element schemes differ in integration: (1) Lobatto four-point, (2) Gauss four-point and (3) Gauss one-point. To compare schemes at the most fundamental level, and identify basic aspects responsible for their behaviours with the varying speed ratio, we analyse 2-D second-order schemes assuming an elastic homogeneous isotropic medium and a uniform grid. We compare structures of the schemes and applied FD approximations. We define (full) local errors in amplitude and polarization in one time step, and normalize them for a unit time. We present results of extensive numerical calculations for wide ranges of values of the speed ratio and a spatial sampling ratio, and the entire range of directions of propagation with respect to the spatial grid. The application of some schemes to real sedimentary basins in general requires considerably finer spatial sampling than usually applied. Consistency in approximating first spatial derivatives appears to be the key factor for the behaviour of a scheme with respect to the P -wave to S -wave speed ratio. [source] The feasibility of electromagnetic gradiometer measurementsGEOPHYSICAL PROSPECTING, Issue 3 2001Daniel Sattel The quantities measured in transient electromagnetic (TEM) surveys are usually either magnetic field components or their time derivatives. Alternatively it might be advantageous to measure the spatial derivatives of these quantities. Such gradiometer measurements are expected to have lower noise levels due to the negative interference of ambient noise recorded by the two receiver coils. Error propagation models are used to compare quantitatively the noise sensitivities of conventional and gradiometer TEM data. To achieve this, eigenvalue decomposition is applied on synthetic data to derive the parameter uncertainties of layered-earth models. The results indicate that near-surface gradient measurements give a superior definition of the shallow conductivity structure, provided noise levels are 20,40 times smaller than those recorded by conventional EM instruments. For a fixed-wing towed-bird gradiometer system to be feasible, a noise reduction factor of at least 50,100 is required. One field test showed that noise reduction factors in excess of 60 are achievable with gradiometer measurements. However, other collected data indicate that the effectiveness of noise reduction can be hampered by the spatial variability of noise such as that encountered in built-up areas. Synthetic data calculated for a vertical plate model confirm the limited depth of detection of vertical gradient data but also indicate some spatial derivatives which offer better lateral resolution than conventional EM data. This high sensitivity to the near-surface conductivity structure suggests the application of EM gradiometers in areas such as environmental and archaeological mapping. [source] Quasi-wavelet solution of diffusion problemsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 12 2004Tang Jiashi Abstract A new method, quasi-wavelet method, is introduced for solving partial differential equations of diffusion which are important to chemical and mechanical engineering. A new scheme for the extension of boundary conditions is proposed. The quasi-wavelet method is utilized to discretize the spatial derivatives, while the Runge,Kutta scheme is employed for the time advancing. The problems of particle diffusion in the electrochemistry reaction and temperature diffusion in plates are studied. Quasi-wavelet solution of the former problem is compared with those of a finite difference method. Solution of the latter problem is calibrated by analytical solution. Numerical results indicate that the quasi-wavelet approach is very robust and efficient for diffusion problems. Copyright © 2004 John Wiley & Sons, Ltd. [source] Development of a class of multiple time-stepping schemes for convection,diffusion equations in two dimensionsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 12 2006R. K. Lin Abstract In this paper we present a class of semi-discretization finite difference schemes for solving the transient convection,diffusion equation in two dimensions. The distinct feature of these scheme developments is to transform the unsteady convection,diffusion (CD) equation to the inhomogeneous steady convection,diffusion-reaction (CDR) equation after using different time-stepping schemes for the time derivative term. For the sake of saving memory, the alternating direction implicit scheme of Peaceman and Rachford is employed so that all calculations can be carried out within the one-dimensional framework. For the sake of increasing accuracy, the exact solution for the one-dimensional CDR equation is employed in the development of each scheme. Therefore, the numerical error is attributed primarily to the temporal approximation for the one-dimensional problem. Development of the proposed time-stepping schemes is rooted in the Taylor series expansion. All higher-order time derivatives are replaced with spatial derivatives through use of the model differential equation under investigation. Spatial derivatives with orders higher than two are not taken into account for retaining the linear production term in the convection,diffusion-reaction differential system. The proposed schemes with second, third and fourth temporal accuracy orders have been theoretically explored by conducting Fourier and dispersion analyses and numerically validated by solving three test problems with analytic solutions. Copyright © 2006 John Wiley & Sons, Ltd. [source] Development of highly accurate interpolation methodfor mesh-free flow simulations III.INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, Issue 7 2003Analysis of accuracy, stability Abstract A highly accurate interpolation method, CIVA, improves the accuracy of mesh-free and grid-less methods by taking into consideration first-order spatial derivatives as variables; an approach based on the same idea as that on which CIP is based. In this study, the accuracy and stability of CIVA is evaluated by analytically and numerically. First, the general formulation of CIVA for the n -dimensional case is described. Since CIVA contains the bubble function, we consider the determination methods: constant curvature condition and utilization of another computing point. Then, the relation between the bubble function in the CIVA method and the accuracy and stability is made clear by the analysis based on the Taylor expansion. Some computations of two-dimensional passive scalar advection and advection,diffusion problems are performed for the verification of accuracy and stability. Copyright © 2003 John Wiley & Sons, Ltd. [source] A new parallelization strategy for solving time-dependent 3D Maxwell equations using a high-order accurate compact implicit scheme,INTERNATIONAL JOURNAL OF NUMERICAL MODELLING: ELECTRONIC NETWORKS, DEVICES AND FIELDS, Issue 5 2006Eugene Kashdan Abstract With progress in computer technology there has been renewed interest in a time-dependent approach to solving Maxwell equations. The commonly used Yee algorithm (an explicit central difference scheme for approximation of spatial derivatives coupled with the Leapfrog scheme for approximation of temporal derivatives) yields only a second-order of accuracy. On the other hand, an increasing number of industrial applications, especially in optic and microwave technology, demands high-order accurate numerical modelling. The standard way to increase accuracy of the finite difference scheme without increasing the differential stencil is to replace a 2nd-order accurate explicit scheme for approximation of spatial derivatives with the 4th-order accurate compact implicit scheme. In general, such a replacement requires additional memory resources and slows the computations. However, the curl-based form of Maxwell equations allows us to construct an effective parallel algorithm with the alternating domain decomposition (ADD) minimizing the communication time. We present a new parallel approach to the solution of three-dimensional time-dependent Maxwell equations and provide a theoretical and experimental analysis of its performance. Copyright © 2006 John Wiley & Sons, Ltd. [source] A fourth-order accurate, Numerov-type, three-point finite-difference discretization of electrochemical reaction-diffusion equations on nonuniform (exponentially expanding) spatial grids in one-dimensional space geometryJOURNAL OF COMPUTATIONAL CHEMISTRY, Issue 12 2004aw K. Bieniasz Abstract The validity for finite-difference electrochemical kinetic simulations, of the extension of the Numerov discretization designed by Chawla and Katti [J Comput Appl Math 1980, 6, 189,196] for the solution of two-point boundary value problems in ordinary differential equations, is examined. The discretization is adapted to systems of time-dependent reaction-diffusion partial differential equations in one-dimensional space geometry, on nonuniform space grids resulting from coordinate transformations. The equations must not involve first spatial derivatives of the unknowns. Relevant discrete formulae are outlined and tested in calculations on two example kinetic models. The models describe potential step chronoamperometry under limiting current conditions, for the catalytic EC, and Reinert-Berg CE reaction mechanisms. Exponentially expanding space grid is used. The discretization considered proves the most accurate and efficient, out of all the three-point finite-difference discretizations on such grids, that have been used thus far in electrochemical kinetics. Therefore, it can be recommended as a method of choice. © 2004 Wiley Periodicals, Inc. J Comput Chem 25: 1515,1521, 2004 [source] Blow up, decay bounds and continuous dependence inequalities for a class of quasilinear parabolic problemsMATHEMATICAL METHODS IN THE APPLIED SCIENCES, Issue 3 2006L. E. Payne Abstract This paper deals with a class of semilinear parabolic problems. In particular, we establish conditions on the data sufficient to guarantee blow up of solution at some finite time, as well as conditions which will insure that the solution exists for all time with exponential decay of the solution and its spatial derivatives. In the case of global existence, we also investigate the continuous dependence of the solution with respect to some data of the problem. Copyright © 2005 John Wiley & Sons, Ltd. [source] Applications of transformed-space non-uniform PSTD (TSNU-PSTD) in scattering analysis without the use of the non-uniform FFTMICROWAVE AND OPTICAL TECHNOLOGY LETTERS, Issue 1 2003Xiaoping Liu Abstract In this work, we extend the transformed-space, non-uniform pseudo-spectral time domain (TSNU-PSTD) Maxwell solver for a 2D scattering analysis. Prior to implementing the PSTD in this analysis, we first transform the non-uniform grids {xi} and {yj} sampled in the real space for describing complex geometries to uniform ones {ui} and {vj}, in order to fit the dimensions of practical structures and utilize the standard fast Fourier transform (FFT). Next, we use a uniform-sampled, standard FFT to represent spatial derivatives in the space domain of (u, v). It is found that this scheme is as efficient as the conventional uniform PSTD with the computational complexity of O(N log N), since the difference is only the factors of du/dx and dv/dy between the conventional PSTD and the TSNU-PSTD technique. Additionally, we apply an anisotropic version of the Berenger's perfectly matched layers (APML) to suppress the wraparound effect at the open boundaries of the computational domain, which is caused by the periodicity of the FFT. We also employ the pure scattered-field formulation and develop a near-to-far-zone field transformation in order to calculate scattered far fields. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 38: 16,21, 2003 [source] Method of lines with boundary elements for 1-D transient diffusion-reaction problemsNUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS, Issue 4 2006P.A. Ramachandran Abstract 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] Peculiarities of soliton motion in molecular systems with high dispersionPHYSICA STATUS SOLIDI (C) - CURRENT TOPICS IN SOLID STATE PHYSICS, Issue 11 2004V. V. Krasilnikov Abstract In this work, features of propagating protons along molecular chain of hydrogen bonds are described from position of soliton dynamics with taking into account interaction of first and second neighbors of a proton sublattice. It is proposed extension of the model that is an endless chain of water molecules in which formation of hydrogen bonds is due to participating one proton of every water molecule, a second proton no participating in hydrogen bond and being confined by covalent bond of an oxygen atom. Nonlinearity is due to peculiar properties of proton sublattice potential. The model used to obtain continual equations which contain the spatial derivatives of the fourth order that is related with dispersion of longwave oscillations. The availability of such a dispersion changes essentially dynamics of the molecular chain, which allows of manifesting new peculiarities of propagating nonlinear excitations. It is shown there are two new sorts of charge density excitations transferred by solitons determined as exact analytic dependences in such a system. (© 2004 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] On the use of the super compact scheme for spatial differencing in numerical models of the atmosphereTHE QUARTERLY JOURNAL OF THE ROYAL METEOROLOGICAL SOCIETY, Issue 609 2005V. Esfahanian Abstract The ,Super Compact Finite-Difference Method' (SCFDM) is applied to spatial differencing of some prototype linear and nonlinear geophysical fluid dynamics problems. An alternative form of the SCFDM relations for spatial derivatives is derived. The sixth-order SCFDM is compared in detail with the conventional fourth-order compact and the second-order centred differencing. For the frequency of linear inertia-gravity waves on different numerical grids (Arakawa's A,E and Randall's Z) related to the Rossby adjustment process, the sixth-order SCFDM shows a substantial improvement on the conventional methods. For the Jacobians involved in vorticity advection by non-divergent flow and in the Bolin,Charney balance equation, a general framework, valid for every finite-difference method, is derived to present the discrete forms of the Jacobians. It is found that the sixth-order SCFDM provides a noticeably more accurate representation of the wave-number distribution of the Jacobians, when compared with the conventional methods. The problem of reconstructing the stream-function field from the vorticity field on a sphere is also considered. For the Rossby,Haurwitz wave, the computation of a normalized global error at different horizontal resolutions in longitude and latitude directions shows that the sixth-order SCFDM can markedly improve on the fourth-order compact. The sixth-order SCFDM is thus proposed as a viable method to improve the accuracy of finite-difference models of the atmosphere. Copyright © 2005 Royal Meteorological Society. [source] Spectral composition of electromagnetic fluctuations induced by a lossy layered systemANNALEN DER PHYSIK, Issue 7-8 2003I. Dorofeyev Abstract We calculate the spectral characteristics of fluctuating electromagnetic fields generated by a half-space covered with a film of an arbitrary thickness. Materials of the half-space and the film are described by different complex permittivities. Expressions for spectral power densities of fluctuating fields and all spatial derivatives expressed via Fresnel coefficients for "p" and "s" waves are derived. Various limiting cases for propagating and evanescent waves in cases of different film thickness are considered. Possible contributions to spectral power densities from interface excitations and wave-guide modes are discussed by analyzing the Fresnel factors in the expressions. Using the results for spatial derivatives a closed analytical expression for a multipolar force acting on a small particle near a half-space is found for multipoles of all orders. The case for a dipole interaction follows directly from a general solution. [source] | |||||||||||||||||||