Finite difference method first derivative

Finite Difference Method To apply the finite difference method, we first spread grid points through the domain. Figure 3-49 shows a uniform mesh of n points (nonuniform meshes are possible, too). The unknown, here c(x), at a grid point x, is assigned the symbol Cj = c(Xi). The finite difference method can be derived easily by using a Taylor expansion of the solution about this point. Expressions for the derivatives are ... 

A finite element method based on these functions would have an error proportional to Ax2. The finite element representations for the first derivative and second derivative are the same as in the finite difference method, but this is not true for other functions or derivatives. With quadratic finite elements, take the region from x,.i and x,tl as one element. Then the interpolation would be... 

Derivative of intensity against structure parameters and thickness can be obtained using the first order perturbation method [31]. The finite difference method can also be used to evaluate the derivatives. Estimates of errors in refined parameters can also be obtained by repeating the measurement. In case of CBED, this can also be done by using different... 

By taking into account the above approximate expressions for the first and second derivatives, the finite-difference method proceeds as follows. First, the... 

The numerical methods for solving differential equations are based on replacing the differential equations by algebraic equations. In the case of the popular finite difference method, this is done by replacing the derivatives by differences. Below we demonstrate this with both first- and second-order derivatives. But first we give a motivational e.xample. 

The model equations were solved numerically by discretizing the partial differential equations (PDEs) with respect to the spatial coordinate (x). Central finite difference formulae were used to approximate the first and second derivatives (e.g. dc,/dx, d77ck). Thus the PDEs were transformed to ODEs with respect to the reaction time and the finite difference method was used in the numerical solution. The recently developed software of Buzzi Ferraris and Manca was used, since it turned out to be more rapid than the classical code of Hindmarsh. 

The numerical solution of the initial-boundary-value problem based on the equation system (44) can be performed (Winkler et al, 1995) by applying a finite-difference method to an equidistant grid in energy U and time t. The discrete form of the equation system (44) is obtained using, on the rectangular grid, second-order-correct centered difference analogues for both distributions f iU, i)/n and f U, t)/n and their partial derivatives of first order. 

All factors on the right-hand side of Eq. (10.3-20) are constant. This equation is a linear first-order difference equation and can be solved by the calculus of finite-difference methods (Gl. M1). The final derived equations are as follows. 

The application of a finite-difference method transforms the first derivatives V[u L)k point Sk into a linear combination of function values y(u L),m at contiguous points Sm around s. This yields two linear difference equations for every inner grid point s. The resulting set of linear equations can be combined into a 2m X 2m matrix equation... 

In molecular dynamics (MD) simulation atoms are moved in space along their lines of force (which are determined from the first derivative of the potential energy function) using finite difference methods [27, 28]. At each time step the evolution of the energy and forces allow the accelerations on each atom to be determined, in turn allowing the atom changes in velocities and positions to be evaluated and hence allows the system clock to move forward, typically in time steps of the order of a few fs. Bulk system properties such as temperature and pressure are easily determined from the atom positions and velocities. As a result simulations can be readily performed at constant temperature and volume (NVT ensemble) or constant temperature and pressure (NpT ensemble). The constant temperature and pressure constraints can be imposed using thermostats and barostat [29-31] in which additional variables are coupled to the system which act to modify the equations of motion. 

In many books, radial flow theory is studied superficially and dismissed after cursory derivation of the log r pressure solution. Here we will consider single-phase radial flow in detail. We will examine analytical formulations that are possible in various physical limits, for different types of liquids and gases, and develop efficient models for time and cost-effective solutions. Steady-state flows of constant density liquids and compressible gases can be solved analytically, and these are considered first. In Examples 6-1 to 6-3, different formulations are presented, solved, and discussed the results are useful in formation evaluation and drilling applications. Then, we introduce finite difference methods for steady and transient flows in a natural, informal, hands-on way, and combine the resulting algorithms with analytical results to provide the foundation for a powerful write it yourself radial flow simulator. Concepts such as explicit versus implicit schemes, von Neumann stability, and truncation error are discussed in a self-contained exposition. 

The basic principle in using the finite difference method to solve BVPs is to replace all the derivatives in the differential equation with difference-quotient approximations. First, the interval a < x < b is discretized into n equally spaced intervals (an unequal spaced interval may also be used) ... 

The basic idea of the finite difference method for solving PDEs is to use a grid in the independent variables and to discretize the PDEs by introdncing difference-quotient approximations, thereby reducing the problem to solving a system of equations. The first derivative in time can be approximated by the forward difference formula. 

The derivative matrix returned by the function deriv.m has the same number of elements as the vector of input data itself. However, it is important to note that, depending on the method of finite difference used, some elements at one or both ends of the derivative vector are evaluated by a different method of differentiation. For example, in first-order differentiation with the forward finite difference metliod with truncation error 0(h), the last element of the returned derivative vector is calculated by backward differences. Another example is the calculation of the second-order derivative of a vector by the central finite difference method with truncation error 0(h ), where the function evaluates tlie first two elements of the vector of derivatives by forward differences and the last two elements of tlie vector of derivatives by backward differences. The reader should pay special attention to the fact that when the function calculates the derivative by the central finite difference method with the truncation error of the order 0(h ), the starting and ending rows of derivative values are calculated by forward and backward finite differences, with truncation error of the order O(h ). 

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