Forward Euler method

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

Equations that arise in modeling the dynamics of homogeneous systems are initial value problems, generally approached with techniques of the Euler type. Initial value problems involve derivatives with respect to time these must be discretized, which can be done using the forward Euler method... 

In these formulas, the subscript refers to the discretized time step, and h is the size of the fixed time step, i.e., A = tk+i - tfe. The difference in the two methods is in the choice of points that are used to estimate the time derivative at the kth point. The discretization used in the forward Euler method will lead to the following approximation of the ODE [where x = f(x)]... 

The forward Euler method is thus referred to as an explicit method, because Xk+i is taken to depend only on points that have been previously calculated, that is, x + depends only on The discretization used in the backward Euler method leads to the following approximation of the ODE... 

For example, the forward Euler method is the simplest one-step algorithm ... 

Figure 2.4 Stability region of the forward Euler method, linear equation to do so ...

For example, the backward Euler is strongly A-stable. Nevertheless, if the value of is calculated by applying iteratively and with a limited number of iterations the relation = y +/ f y, j,r +i, where y, +i was calculated using the forward Euler method, the method can become unstable. 

To solve the equations of motion, various integration methods could be used as an example here, the forward Euler method is presented for quantification of velocity and positions of the particles at different times ... 

We can obtain an initial guess, zq, by using the forward Euler method, yi=yo-1 — 0.1 P = 0.9. The first iteration with the Newton method gives... 

This includes as special cases the explicit (forward) Euler method for 0 = 0, the implicit (backward) Euler method for 0 = 1, and the Crank-Nicholson method for 0 = 1/2. 

My first attempt to calculate the time history of a geochemical system (Section 2.3) used the obvious approach (the direct Euler method) of evaluating the time derivatives and stepping forward. But it was not sue-... 

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

Forward differences, first order explicit Euler method ... 

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... 

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... 

Sometimes the ODEs that arise in studies in nonlinear dynamics can be solved using explicit methods (such as the forward Euler) which require less computations per step and are thus cheaper and ter to implement. The Runge-Kutta femily of algorithms are a popular implementation of the explicit methods. Runge—Kutta methods begin with a Taylor series expansion the order of the particular Runge-Kutta method used is simply the highest order term retained in the Taylor series. 

The two equations, [53] and [55], form a system of coupled ODEs with the variable z playing the role of the independent variable. Given initial conditions at a point Zq these equations can be solved by standard numerical routines such as those discussed in the previous section. Because much computational effort is required to evaluate each p, at each increment of the independent variable z, a method that does not require too many evaluations of the right hand side of the iterative equation is desirable. Usually, a simple forward Euler routine is quite adequate for these purposes. If a multistep algorithm is used, the Adams-Bashforth method has been recommended by Kubicek and Marek the first-order Adams-Bashforth algorithm is, in fact, equivalent to the simple forward Euler algorithm. 

As a simple illustration, consider the harmonic oscillator with Hamiltonian H(q,p) = p /2 - - (ill and the invariant distributions obtained using several numerical methods. Applying Forward Euler to the harmonic oscillator, we have... 

All the terms in the right-hand side of Eq. 12.137 are known, and hence, this makes the forward difference in time rather attractive. However, as in the Euler method (Chapter 7), this forward difference scheme in time suffers the same handicap, that is, it is unstable if the grid size is not properly chosen. Using the stability analysis (von Rosenberg 1969), the criterion for stability (see Problem 12.12) is... 

A MatLab code was created to solve equations 2 to 6 simultaneously. The constants in table 1 were used to generate a solution and the numerical methods apphed are Euler method and finite difference method. The finite difference is used to estimate the second order space differential and the forward Euler... 

In a forward-marching method such as Euler s method, we are more interested in the total error propagation over multiple usage of the algorithm than in the local one-step error. If we let Cj be the error between the approximate solution, rcj, and exact solution, x ti) to the differential equation, then... 

Step 5. A valne for 0 at the next moment (at the next time-line ) follows from Euler Forward s method based on Eqnation 15.14 0, = + 7,, At A/v ... 

Suppose now that we do not have the equation, but we have the experiment itself. We can fill up the stirred reactor with reactant at concentration Cq, run it for some time, and record the time series of c(t). Using the results of a short run (over, say, one minute), we can now estimate the slope, dc/dt at t=0, and predict (using the Euler method) where the concentration will be in, say 10 minutes. Now, instead of waiting nine minutes for the reactor to get there, we stop the experiment and immediately start a new one. We reinitialize the reactor at the predicted concentration, run for one more minute, and use forward Euler to predict what the concentration will be 20 minutes down the line. We are substituting short, appropriately initialized experiments, and estimation based on the experimental results, for the function evaluations that the subroutine with the closed form f(c) would return. We are in effect doing forward Euler again, but the coefficients of the local linear model are obtained using experimentation on demand (Cybenko, 1996) rather than function evaluations of an n priori available model. 

In contrast to the forward (explicit) Euler method, which uses the slope at the left-hand side to step across the interval, the imphcit verison of the Euler method crosses the interval by using the slope at the right-hand side, as shown in Figure 6.8. The implicit formula does not give any direct approximation ofy +i, instead an iterative method, e.g. the Newton method, is added inside the loop, thus advancing the differential equation to solve fory +1. This obviously comes at the price of more computation, but allows stability... 

One of the earliest techniques developed for the solution of ordinary differential equations is the Euler method. This is simply obtained by recognizing that the left side of Eq. (5.55) is the first forward finite difference of y at position v. 

The accuracy of the Euler method can be improved by utilizing a combination of forward and backward differences. Note that the first forward difference of y at i is equal to the first backward difference of y at (i + 1) ... 

The simplest numerical method for solving one ODE is called the Euler method and is based on approximating the derivative with a forward difference approximation to the derivative as follows ... 

This method applies a centered difference approximation for the derivative and an average value for / on the right-hand side. The finite difference analog is centered about the point. That is, the differential equation is discretized at and f y + i2, t +in) is approximated by the average of the end-point values. Therefore, this method is an average of the forward and backward Euler methods, and the discrete approximation appears in the following equation ... 

Solution by centered difference in x, forward difference in 0, is now illnstrated. This approach suffers the same shortcomings as the Euler method for ODEs (the truncation error is C2(Ax)). By replacing the derivatives with the appropriate finite difference analogs (see Table 4.2) as follows ... 

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... 

Both methods are first order in time but for practical purposes the explicit Euler is the easiest to apply due to the fact that the only unknown is the value 4>k+l all other terms are evaluated in the Tcth time step, and due to prescribed initial conditions are always known. Hence, the value of cj>k+1 can easily be solved for using eqn. (8.74), by marching forward in time. In the implicit Euler case, the whole right hand side of the equation is evaluated in the future, and must therefore be generated and solved for every time step. When marching... 

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