Euler s method

C. S. Peskin and T. Schlick. Molecular dynamics by the backwsird Euler s method. Comm. Pure App. Math., 42 1001-1031, 1989. 

Runge-Kutta methods are explicit methods that use several function evaluations for each time step. Runge-Kutta methods are traditionally written for/(f, y). The first-order Runge-Kutta method is Euler s method. A second-order Runge-Kutta method is... 

This set of ordinaiy differential equations can be solved using any of the standard methods, and the stability of the integration of these equations is governed by the largest eigenvalue of AA. If Euler s method is used for integration, the time step is hmited by... 

Equation (3-159) is the basic relationship of this method. Several techniques have been developed for the estimation of AylAx. The simplest of these, known as Euler s method, is to evaluate Ay/Ax at jcq. From Eq. (3-156), this gives (A v/Ajc)o = kyoi which, used in Eq. (3-159), yields... 

If y, is given as the initial value, y, can be computed by Euler s method, or more accurately as... 

This form is a closed type formula since it does not allow direct steps from x, to x.,, but uses the basic Euler s method to estimate y j, thus... 

The first result, for At = 0.5, shows a negative result for a(t ax) due to the very large value for At. For smaller values of At, the calculated values for a tmax) are physically realistic and the errors decrease by roughly a factor of 4 as the time step decreases by a factor of 4. Thus, the error is proportional to At. Euler s method is said to converge order At, denoted 0(At). 

Convergence order At for Euler s method is based on more than the empirical observation in Example 2.4. The order of convergence springs directly from the way in which the derivatives in Equations (2.11) are calculated. The simplest approximation of a first derivative is... 

V. The auxiliary equation is normally an algebraic equation rather than an ODE. In chemical engineering problems, it will usually be an equation of state, such as the ideal gas law. In any case, the set of ODEs can be integrated numerically starting with known initial conditions, and V can be calculated and updated as necessary. Using Euler s method, V is determined at each time step... 

In this chapter we described Euler s method for solving sets of ordinary differential equations. The method is extremely simple from a conceptual and programming viewpoint. It is computationally inefficient in the sense that a great many arithmetic operations are necessary to produce accurate solutions. More efficient techniques should be used when the same set of equations is to be solved many times, as in optimization studies. One such technique, fourth-order Runge-Kutta, has proved very popular and can be generally recommended for all but very stiff sets of first-order ordinary differential equations. The set of equations to be solved is... 

Computational Scheme for Gas-Phase PFRs. A general procedure for solving the reactor design equations for a piston flow reactor using the marching-ahead technique (Euler s method) has seven steps ... 

Equation (8.22) for a(0) is also special because, due to symmetry, there is only one adjacent point, a(l). The overall set may be solved by any desired method. Euler s method is discussed below and is illustrated in Example 8.5. There are a great variety of commercial and freeware packages available for solving simultaneous ODEs. Most of them even work. Packages designed for stiff equations are best. The stiffness arises from the fact that VJJ) becomes very small near the tube waU. There are also software packages that will handle the discretization automatically. 

Euler s method for solving the above set of ODEs uses a first-order, forward difference approximation in the -direction. Equation (8.16). Substituting this into Equation (8.21) and solving for the forward point gives... 

Example 8.5 Use the method of lines combined with Euler s method to determine the mixing-cup average outlet for the reactor of Example 8.4. ... 

Flow in a Slit. Turning to a slit geometry, a flat velocity profile gives the simplest possible solution using Euler s method. The stability limit is independent of y ... 

A first-order difference approximation for the axial derivative, 9(/0 z)/9j, is consistent with the first-order convergence of Euler s method. The convected-mean concentration is calculated from the dimensionless version of Equation (8.4) ... 

The overall solution is based on the method of lines discussed in Chapter 8. The resulting DDEs can then be solved by any convenient method. Appendix 13.2 gives an Excel macro that solves the DDEs using Euler s method. Figure 13.9 shows the behavior of the streamlines. 

The integration of the state equations (Equation 10.21) by the fully implicit Euler s method is based on the iterative determination of x(t1+i). Thus, having x(t,) we solve the following difference equation for x(t, i). 

Figure 10.1 Schematic diagram of the sequential solution of model and sensitivity equations. The order is shown for a three parameter problem. Steps l, 5 and 9 involve iterative solution that requires a matrix inversion at each iteration of the fully implicit Euler s method. All other steps (i.e., the integration of the sensitivity equations) involve only one matrix multiplication each.

If Euler s method is used for integration, the time step is limited by... 

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