Implicit Euler method

Implicit Methods By using different interpolation formulas involving y, it is possible to cferive imphcit integration methods. Implicit methods result in a nonhnear equation to be solved for y so that iterative methods must be used. The backward Euler method is a first-order method. 

Once all of the conditions were determined and parameters chosen, the equations were solved by an implicit Euler method. The program was written with a self adjusting step size and analytic Jacobian to reduce error and run time. 

The truncation errors in (5.9) and (5.12) are of the same magnitude, but the implicit Euler method (5.11) is stable at any positive step size h. This conclusion is rather general, and the implicit methods have improved stability properties for a large class of differential equations. The price we have to pay for stability is the need for solving a set of generally nonlinear algebraic equations in each step. 

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... 

The formulas (5.7) and (5.11) of explixit and implicit Euler methods, respectively, are unsymmetrical, using derivative information only at one end of the time interval of interest. Averaging the slopes of the two tangent lines means using more information, and gives... 

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.

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 . 

As discussed in Section 15.3.2 on the implicit solution of transient differential equations, one step of the backward Euler method takes the form... 

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

Using a stepsize of h = 10-2 s integrate the equation set with the implicit Euler method over the range t = 0 to 5 s. Plot the log of species concentration versus time for all three species. 

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. 

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]. 

The system of equations is discretized in space by a finite voJume approach, while for the time integration an implicit Euler method is used. Particle and flow model are solved consecutively, which implies that conditions in the bed change slowly compared to the integration step. To reduce the required computation time the flow model is solved here only for one dimension even if the software library TOSCA provides also classes for a two dimensional approach. 

Backward differences, first order implicit Euler method ... 

If the integral on the right side of (12.55) is estimated using the value of the integrand at the final point, we obtain the first order implicit or backward Euler Method ... 

If data at tn+i is included in the interpolation polynomial, implicit methods, known as Adams-Moulton methods, are obtained. The first order method coincides with the implicit Euler method, and the second order method coincides with the trapezoid rule. The third order method is written as ... 

Implicit methods have been developed to overcome this problem. The backward Euler method is... 

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. 

This algorithm is the backward Euler ox fully implicit Euler method. Because this formula is implicit, a system of algebraic equations must be solved to calculate c"+l from cf for i — 1,2,..., N. This step is expensive because the Jacobian of the ODE system needs to be inverted, a process that requires on the order of N3 operations. This inversion should in principle be repeated in each step. 

Because Xk+i appears on both sides of this equation, additional steps are required to solve for x +i before the approximation can be used to calculate it. (This can be done via iteration, such as through the Newton-Raphson method.) Hence, the backward Euler method is also referred to as an implicit method. The trapezoidal algorithm averages the information from the forward and backward Euler algorithms such that the iteration equation to be used is... 

Lee and Dudukovic [18] described an NEQ model for homogeneous RD in tray columns. The Maxwell-Stefan equations are used to describe interphase transport, with the AIChE correlations used for the binary (Maxwell-Stefan) mass-transfer coefficients. Newton s method and homotopy continuation are used to solve the model equations. Close agreement between the predictions of EQ and NEQ models were found only when the tray efficiency could correctly be predicted for the EQ model. In a subsequent paper Lee and Dudukovic [19] presented a dynamic NEQ model of RD in tray columns. The DAE equations were solved by use of an implicit Euler method combined with homotopy continuation. Murphree efficiencies calculated from the results of an NEQ simulation of the production of ethyl acetate were not constant with time. 

Other examples of implicit Runge-Kutta algorithms include the backward Euler methods ... 

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