Euler algorithm

For example, the SHAKE algorithm [17] freezes out particular motions, such as bond stretching, using holonomic constraints. One of the differences between SHAKE and the present approach is that in SHAKE we have to know in advance the identity of the fast modes. No such restriction is imposed in the present investigation. Another related algorithm is the Backward Euler approach [18], in which a Langevin equation is solved and the slow modes are constantly cooled down. However, the Backward Euler scheme employs an initial value solver of the differential equation and therefore the increase in step size is limited. 

To check the effect of integration, the following algorithms were tried Euler, explicit Runge-Kutta, semi-implicit and implicit Runge-Kutta with stepwise adjustment. All gave essentially identical results. In most cases, equations do not get stiff before the onset of temperature runaway. Above that, results are not interesting since tubular reactors should not be... 

Discretization error depends on the step size, i.e., if Ax. —> 0, the algorithm would theoretically be exact. The error for Euler method at step N is 0 N(Ax) and total accumulated error is 0 (Ax), that is, it is a first-order 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. 

Let me make my own personal preference clear from the outset. I have solved literally hundreds of systems of ODEs for chemical engineering systems over my 30 years of experience, and t have found only one or two situations where the plain old simple-minded first-order Euler algorithm was not the best choice for the problem. We will show some comparisons of different types of algorithms on different problems in this chapter and the next. 

A. EULER ALGORITHM. The simplest possible numerical-integration scheme (and the most useful) is Euler (pronounced oiler ), illustrated in Fig. 4.7. Assume we wish to solve the ODE... 

The explicit first-order Euler algorithm is used. The variables that we are solving for as functions of time are V and H. The right-hand sides of Eqs. (5.3) and (5.4) are the derivative functions. These are called VDOT and HDOT in the program. At the nth step in time... 

The new values of H and V at the (n + l)st step are calculated from the Euler algorithm with a step size of DELTA. 

The Stratonovich SDEs for either generalized or Cartesian coordinates could be numerically simulated by implementing the midstep algorithm of Eq. (2.238). Evaluation of the required drift velocities would, however, require the evaluation of sums of derivatives of B or whose values will depend on the decomposition of the mobility used to dehne these quantities. This provides a worse starting point for numerical simulation than the forward Euler algorithm interpretation. 

To explain the stability characteristics of the forward Euler algorithm, consider the following model problem [215] ... 

By adding the forward (explicit) finite-difference approximation to each side of this equation, we can identify both the explicit Euler algorithm and an expression for the local truncation error ... 

The first three terms represent the forward Euler algorithm operating on the exact solution, with the last term [in square brackets] providing a measure of the local truncation error. The local truncation error can be identified through a Taylor series expansion of the solution about the time tn ... 

To advance the solution from yn to yn+ (Eq. 15.12), the forward Euler algorithm extrapolates the slope of the numerical solution at tn to tn+. Since there is always some error in the numerical solution, the slope evaluation is computed from one of the nearby solutions (i.e., one that originated from a different initial condition). 

The first three terms represent the implicit Euler algorithm and the remaining [bracketed] term represents the local truncation error. A Taylor series expansion about tn+ (in the negative t direction) yields an expression for y(t )... 

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