Explicit Euler method

For simple systems, the McDowell molecular-orbital technique would seem to be more time-consuming than that of SJG. In more complicated situations, however, this approach should lead to more accurate results, not only by using a Runge-Kutta rather than Euler method, but also by employing directly Ps(e), rather than its Fourier transform Gi(tX whose explicit form may not be known. 

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... 

To compare the explicit and implicit Euler methods we exploited that the solution (5.3) of (5.2) is known. We can, however, estimate the truncation error without such artificial information. Considering the truncated Taylor series of the solution, for the explicit Euler method (5.7) we have... 

As stated above, the spatial derivative is approximated without regard to the time level. The distinction between explicit and implicit solutions depends on the time level at which the spatial derivatives are evaluated. Finite-difference stencils for explicit and implicit Euler methods are illustrated in Fig. 4.13. 

Fig. 4.13 Finite-difference stencils for the explicit and implicit Euler methods. The spatial index is j and the time index is n. For equally spaced radial mesh intervals of dr, rj = (j — 1 )dr, 1 < j < J. For equally spaced time intervals, tn = (n — 1 )dt, n > 1.

The explicit (or forward) Euler method begins by approximating the time derivative with a first-order finite difference as... 

Fig. 15.3 Illustration of a stable and unstable solution to the model problem (Eq. 15.5) by the forward (explicit) Euler method.

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. 

Implicit methods, which have far better stability properties than explicit methods, provide the computational approach to solving stiff problems. The simplest implicit method is the backward (implicit) Euler method, which is stated as... 

Compared to the explicit Euler method (Eq. 15.9), note that the right-hand side is evaluated at the advanced time level tn+1- If f(t, ) is nonlinear then Eq. 15.22 must be solved iteratively to determine yn+. Despite this complication, the benefit of the implicit method lies in its excellent stability properties. The lower panel of Fig. 15.2 illustrates a graphical construction of the method. Note that the slope of the straight line between y +i and yn is tangent to the nearby solution at tn+, whereas in the explicit method (center panel) the slope is tangent to the nearby solution at t . 

The explicit Euler method stability criterion was given as... 

For a step size of h = 10 5 s evaluate the left-hand side of this equation for species A, B, and C using the A s found in step 3 above. Is the explicit Euler method predicted to be stable for this stepsize hi... 

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. 

Various integration methods were tested on the dynamic model equations. They included an implicit iterative multistep method, an implicit Euler/modified Euler method, an implicit midpoint averaging method, and a modified divided difference form of the variable-order/variable-step Adams PECE formulas with local extrapolation. However, the best integrator for our system of equations turned out to be the variable-step fifth-order Runge-Kutta-Fehlberg method. This explicit method was used for all of the calculations presented here. 

D APC ASTEEM BC CAM3 Three-dimensional The analytical predictor of condensation The adaptive step time-split explicit Euler method Black carbon The community atmospheric model v. 3 ... 

Forward differences, first order explicit Euler method ... 

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 ... 

The formula is known both as the improved Euler formula and as Heun s method [170]. This method has roughly the stability of the explicit Euler method. Unfortunately, the stability may not be improved by iterating the corrector because this iteration procedure converges to the trapezoid rule solution only if At is small enough. 

In order to construct higher-order approximations one must use information at more points. The group of multistep methods, called the Adams methods, are derived by fitting a polynomial to the derivatives at a number of points in time. If a Lagrange polynomial is fit to /(t TO, V )i. .., f tn, explicit method of order m- -1. Methods of this tirpe are called Adams-Bashforth methods. It is noted that only the lower order methods are used for the purpose of solving partial differential equations. The first order method coincides with the explicit Euler method, the second order method is defined by ... 

For simplicity, the momentum equation (12.153) is solved with the explicit Euler method in time. Then, we obtain ... 

Methods similar to this are commonly used to solve the momentum equation when an accurate time history of the flow is required. In these particular cases, more accurate time advancement methods than the first order Euler method must be used. Explicit methods are preferred for (fast) transient flows because... 

While the examples given here have dealt only with chemical reaction kinetics, the method illustrates how one can, in general, solve single as well as coupled differential equations. Euler s explicit method is useful as a qualitative tool it is easily implemented, and can provide a reasonably close result when Af is sufficiently small. The latter requirement, however, may make the explicit method impractical on a spreadsheet. For quantitative work, an implicit method is usually required, as it provides a better approximation given the limited number of iterations practical on a spreadsheet. 

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. 

FIGURE 25.9 Numerical solution of the stiff ODE example using the explicit Euler method and At = 0.001. Also shown is the true solution. 

Even after the use of the PSSA the remaining problem is stiff and its integration cannot be performed efficiently with an explicit Euler method. Fully implicit, stiffly stable integration techniques have been developed and are routinely used for such problems. 

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