Numerical methods Euler method

Eq. 4.38 with its corresponding boundary conditions can be solved by numerical methods (Euler s, Runge-Kutta. ..). This can be done now rather easily by using available software. Numerical solution of Eq. 4.38 is obtained as a set of data of P for different values of the independent variable z and the parameter O, that can be... 

The method of Ishida et al [84] includes a minimization in the direction in which the path curves, i.e. along (g/ g -g / gj), where g and g are the gradient at the begiiming and the end of an Euler step. This teclmique, called the stabilized Euler method, perfomis much better than the simple Euler method but may become numerically unstable for very small steps. Several other methods, based on higher-order integrators for differential equations, have been proposed [85, 86]. 

Solution of Batch-Mill Equations In general, the grinding equation can be solved by numerical methods—for example, the Luler technique (Austin and Gardner, l.st Furopean Symposium on Size Reduction, 1962) or the Runge-Kutta technique. The matrix method is a particiilarly convenient fornmlation of the Euler technique. 

The results are shown in Figure 2-3, in which the solid line is the exact solution. This numerical approach shows no sign of instability even for a time step of 40 years, nearly five times larger than the residence time of atmospheric carbon dioxide (distime). In fact, the reverse Euler method is nearly always stable, and so I shall use it from now on. 

Fig, 2-3. The recovery of atmospheric carbon dioxide calculated by the reverse Euler method. The solid line is the analytical solution, and the lines with markers show the numerical results calculated with program DGC02 using time steps delx = 10, 20, and 40 years. The numerical solution is stable for all time steps. 

More complicated numerical methods, such as the Runge-Kutta method, yield more accurate solutions, and for precisely formulated problems requiring accurate solutions these methods are helpful. Examples of such problems are the evolution of planetary orbits or the propagation of seismic waves. But the more accurate numerical methods are much harder to understand and to implement than is the reverse Euler method. In the following chapters, therefore, I shall show the wide range of interesting environmental simulations that are possible with simple numerical methods. 

The previous chapter showed how the reverse Euler method can be used to solve numerically an ordinary first-order linear differential equation. Most problems in geochemical dynamics involve systems of coupled equations describing related properties of the environment in a number of different reservoirs. In this chapter I shall show how such coupled systems may be treated. I consider first a steady-state situation that yields a system of coupled linear algebraic equations. Such a system can readily be solved by a method called Gaussian elimination and back substitution. I shall present a subroutine, GAUSS, that implements this method. 

Numerical integration of sets of differential equations is a well developed field of numerical analysis, e.g. most engineering problems involve differential equations. Here, we only give a very brief introduction to numerical integration. We start with the Euler method, proceed to the much more useful Runge-Kutta algorithm and finally demonstrate the use of the routines that are part of the Matlab package. 

Although not recommended for practical use, the classical Euler extrapolation is a convenient example to illustrate the basic ideas and problems of numerical methods. Given a point (tj y1) of the numerical solution and a step size h, the explicit Euler method is based on the approximation (yi+1 - /( i+l " 4 dy/dt to extrapolate the solution... 

Figure 15.3 illustrates the performance of the explicit Euler method on the model problem, Eq. 15.5. In both panels, the time step is h = 0.1, but the left-hand panel has X = 10 and the right-hand panel has X = 30. The heavy lines show the y = t2 + 1 solution and the course of the numerical solution. The thinner lines show solution trajectories from different initial conditions. 

The PES in the vicinity of IRC is approximated by an (N - -dimensional parabolic valley, whose parameters are determined by using the gradient method. Specific numerical schemes taking into account p previous steps to determine the (p + l)th step render the Euler method stable and allow one to optimize the integration step in Eq. (8.5) [Schmidt et al., 1985]. When the IRC is found, the changes of transverse normal vibration frequencies along this reaction path are represented as... 

Depending on the numerical techniques available for integration of the model equations, model reformulation or simplified version of the original model has always been the first step. In the Sixties and Seventies simplified models as sets of ordinary differential equations (ODEs) were developed. Explicit Euler method or explicit Runge-Kutta method (Huckaba and Danly, 1960 Domenech and Enjalbert, 1981 Coward, 1967 Robinson, 1969, 1970 etc) were used to integrate such model equations. The ODE models ignored column holdup and therefore non-stiff integration techniques were suitable for those models. 

The numerical model-Simulator NV-Simulator V. At this point, we must find the more suitable variant for passing from the differential or partly differential model equations to the numerical state. For the case of the monodimensional model, we can select the simplest numerical method - the Euler method. In order to have a stable integration, an acceptable value of the integration time increment is recommended. In a general case, a differential equations system given by relations (3.55)-(3.56) accepts a simple numerical integration expressed by the recurrent relations (3.57) ... 

For systems involving recycle streams or intermediate feed locations, the method of successive substitution can be used [Mohan and Govind, 1988a]. Moreover, multiple reactions including side reactions and series, parallel or series-parallel reactions result in strongly coupled differential equations. They have been solved numerically using an implicit Euler method [Bernstein and Lund, 1993]. 

Conversion for maximum mixedness. The Euler method will be used for numerical integration ... 

The standard Euler methods and Runge-Kutta methods do not converge for stiff ODE S. A still system can be defined as one in which the stability of the numerical methods used becomes an issue. Maple has an inbuilt stiff solver. 

This method is more accurate than the Euler method, in the sense that it tends to make a smaller error = x(f )-x for a given step size At. In both cases, the error —> 0 as Az —> 0, but the error decreases faster for the improved Euler method. One can show that E = Az for the Euler method, but E (Az) for the improved Euler method (Exercises 2.8.7 and 2.8.8). In the jargon of numerical analysis, the Euler method is first order, whereas the improved Euler method is second order. 

Numerous predictor-corrector methods have been developed over the years. A simple second order predictor-corrector method can be constructed by first approximating the solution at the new time step using the first order explicit Euler method ... 

There are two methods of obtaining a curve of r vs x from Eqs. (B) and (D). The first approach is to write Eq. (B) in difference form for a small change in conversion. Ax, and solve by stepwise numerical integration. As an illustration let us follow through three incremental calculations using the modified Euler method. We write Eq. (B) as... 

The performance of numerical methods for chemical continuity equations is generally characterized in terms of accuracy, stability, degree of mass conservation, and computational efficiency. The simplest of such methods is provided by the forward Euler or fully explicit scheme, by which the solution y" 1 at time tn y is given by... 

The computer code for the electrode equilibration model (EHDRIFT) was written in PASCAL for use on a microcomputer. The program calculates the rest potential which is the EMF value where the currents sum to zero. If the system is in homogeneous equilibrium the rest potential will represent the system Eh. The numerical algorithm uses Eulers method (il) to integrate Equation 2, which involves recalculation of the aqueous concentrations (Equations 4 and 5) at each time step. A full listing of the source code can be found in Kempton (12). 

The term I = I t) describes the additional bromide production that is induced by external illumination [32]. Below all numerical simulations with the Oregonator model are performed using an explicit Euler method on a 380 X 380 array with a grid spacing Ax = 0.14 and time steps At = 0.002. 

Usually Eq. (29.12) is solved by the Euler method of numerical approximation, in which the changes in all fractions during successively short time intervals At (say, 30 s) are calculated by the approximation dxjdt = AxJAt. Changes in S and AB u with screen size and (if known) with time can be incorporated. A computer is needed to make the lengthy calculations. The method is illustrated in the following example. 

If a mathematical method for solving a differential equation cannot be found, numerical methods exist for generating numerical solutions to any desired degree of accuracy. Euler s method and the Runge-Kutta method were presented. 

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