Euler's Method
Euler's Method is a numerical technique used to approximate the solutions of ordinary differential equations. It involves breaking down the differential equation into small steps and using the tangent line at each step to approximate the next value. While it is a simple and intuitive method, it may not always provide highly accurate results, especially for complex equations.
5 Key excerpts on "Euler's Method"
eBook - ePubFundamentals of Vehicle Dynamics and Modelling
A Textbook for Engineers With Illustrations and Examples
- Bruce P. Minaker(Author)
- 2019(Publication Date)
- Wiley(Publisher)
...For example, setting,,,,, gives Euler's Method, the simplest numerical method for solving an ODE. (4) The subscript again refers to the iteration number, but unlike the Newton–Raphson algorithm, with each application of Euler's Method, time is incremented by the stepsize. The method progresses by approximating the curved true solution by a straight line of equal slope at the point. This slope is multiplied by the time step to find the change in the dependent variable, which is then added to the initial value. Of course, Euler's Method does not give the true solution, unless the true solution also happens to be a straight line, but if the step size is chosen sufficiently small, then the numerical solution will be a good approximation to the true solution. While shrinking the step size will force the numerical solution to converge to the true solution, it also increases the number of steps required, and therefore the computational effort. The optimal step size to balance accuracy and effort is problem‐dependent, and often requires some experimentation to find a suitable value. While illustrative, Euler's Method is rarely used in practice, as there are many other methods that provide a better trade‐off between accuracy and effort. The multi‐step methods generally fall into one of three important categories: Adams–Bashforth, Adams–Moulton, and Gear, sometimes referred to as Backwards Differentiation Formulae (BDF). The Adams‐Bashforth methods are explicit methods, meaning that the desired unknown can be isolated and solved. The Adams–Moulton and Gear methods are implicit, meaning that the solution for depends on, which in turn depends on, so a recursion is required at each step to advance the solution...
eBook - ePub- Andrés Illanes, Lorena Wilson, Carlos Vera(Authors)
- 2013(Publication Date)
- Wiley(Publisher)
...However, interest is often focused on the value of the property at the end of the process, and for that purpose it is necessary to solve the differential equations, which is not always an easy task as they can be too complex or have no analytical solution. Numerical methods allow these problems to be overcome, offering various procedures by which to determine the value of the property for a given value of the independent variable. This appendix presents the simplest techniques for solving differential equations numerically. The reader who often deals with this kind of problem is recommended to consult a dedicated textbook, such as Chapra and Canale [1]. Two of the classic methods for solving ODEs with defined initial conditions—Euler and Runge–Kutta—will be presented first, followed by some techniques for solving ODEs defined by a boundary condition, including the finite-difference method in particular. A.4.1 The Euler Method Consider the following problem to be solved numerically: (A.48) From Equation A.3, it is possible to linearize y around a small interval using Taylor's series [x 0, x]. Replacing Equation A.48 into Equation A.3 gives: (A.89) In this manner, the trajectory of y can be calculated in an iterative way using a small integration step (h) from an initial value. (A.90) (A.91) The main advantage of this method is its simplicity. Nevertheless, it has important drawbacks: the global error in its prediction is highly dependent on h and the method is conditionally stable. The error can be decreased by reducing h. However, values of h lower than 10 −8 are not recommended in most cases. In the same way, the method improves its stability at lower values of h. A.4.2 The Fourth-Order Runge–Kutta Method This method is one of the most popular numerical techniques for solving ODEs; in fact, in several software packages it is considered the default option, due to its low error and relative simplicity...
eBook - ePubHydraulic Modelling: An Introduction
Principles, Methods and Applications
- Pavel Novak, Vincent Guinot, Alan Jeffrey, Dominic E. Reeve(Authors)
- 2018(Publication Date)
- CRC Press(Publisher)
...equation d y d x = f (x, y) over the interval a ≤ x ≤ b using n steps, each with length h, subject to the initial condition y (a) = y 0. Method : Let y r be the numerical approximation to the exact solution y (x r). As n equal-length steps are to be used, it follows that h = (b − a)/ n, and so x r = a + rh, with r =0,1,2, …, n. Then the value y r +1 is given in terms of r and h by y r +1 = y r + hf (x r,y r) for r = 0,1,2, …, n −1. It can be shown that the Euler method is accurate to the order of h, in the sense that the error will be approximately h, and this is shown by saying that the accuracy is O (h). Although this accuracy can be improved by reducing the value of h, this will result in more steps when advancing a calculation over an interval a < x < b, with a corresponding build-up of truncation errors. Consequently, if greater accuracy is required, a different method of integration must be used. In the Euler method the accuracy O (h) is approximated by h raised to the power 1, and because of this the Euler method is called a first-order method of integration. The Euler method is an example of a one-step method, because the calculation of y r +1 is uniquely determined by the value of y r at the previous step in the calculation. Although the Euler algorithm as given here involves a sequence of steps, all with the same increment h in x, if necessary the step length can be changed at any stage in the calculation. There are many accurate finite-difference methods of integration based on the representation of derivatives by difference quotients. However, instead of discussing these, the highly versatile fourth-order Runge–Kutta method will be described, and its modification to form the self-adaptive Runge–Kutta–Fehlberg method will be mentioned...
- Alexey A. Voinov(Author)
- 2010(Publication Date)
- Academic Press(Publisher)
...These equations are then solved numerically, using a numerical method on a computer. The simplest numerical method is given by the approximation that is used in (3.8). The equation in (3.8) is called a difference equation, and it is a numerical approximation of a differential equation. As we have seen above, such difference equations are discrete and can be solved on a computer by going through all the time-steps starting from the initial condition. The equation in (3.8) is also called the Euler method, which is the simplest way to find a numerical solution for a differential equation. This method also creates quite large errors, especially when the right-hand side is changing fast. There are also other numerical methods that are way more efficient than Euler’s method. One of the most widely used methods is the Runge–Kutta method. This method also takes the value of x at time t and calculates the value of x at time t + dt. However, in the simple version of this method, instead of assuming that the transition function f (t, x) is the same over the whole period dt, as in Figure 3.2, we approximate the value of the function at mid-point t + dt /2 and use this value to improve the result. Instead of the equation we use the equation: where k 1 = f (t, x (t)) dt. Note that here we jump to the next time point at t + dt using the forecasted value at mid-point rather than the value at the beginning point, t. Also note that any improvement in accuracy in this method comes at a price: we have to calculate the transition function f twice to make the move to the next time-step. Why not run the Euler method with halved time-step, instead? The amount of calculations will be about the same: same two calculations of the transition function. The formula would be: Substituting, we get This is pretty close to the above, although not exactly the same...
- Bernard Liengme(Author)
- 2008(Publication Date)
- Academic Press(Publisher)
...For an initial value problem we need to know the initial value of the two values of the dependent variables. Alternatively, the problem may be defined by specifying some conditions at one value of x and others at another value of x. Such problems are called boundary value problems. Exercise 1: Euler's Method Euler developed a method for finding the approximate solution to initial value problems. Let the differential equation to be solved have the form of Equation 14.5 and let the initial value of y be y 0. (14.5) Let the solution (i.e., the integral of Equation 14.5) have the form of Equation 14.6. (14.6) Consider the two curves in Figure 14.1 where f(x) is the differential to be solved and g(x) is the integrated solution. Figure 14.1 From Equation 14.5, we may calculate any value y′ 1, y′ 2, …, y′ n. We already know the value of y 0 – the initial value. Our task is to find values for y 1, y 2, …, y n. Integration of Equation 14.5 from x 0 to x 1 yields Equation 14.7. (14.7) The second term on the right is the area under the curve f(x,y) between the two x -values. Euler approximated this to the area of the rectangle defined by y′ 0, y′ 1, x 0, and x 1. The approximate value of y 1 is then given by Equation 14.8, or by Equation 14.9 when the x increment is represented by h. (14.8) (14.9) Having found an approximation for y 1, we may now find an approximation for y 2 (14.10) In general, the value of the approximation at one point is found from the previous one using (14.11) In this Exercise, Euler's Method is used to find an approximate solution of the differential equation dy/dx = xy, with the initial value y (0) = 1. The approximation is compared to the analytical solution y = exp(x 2 /2). (a) Open a new workbook. Enter the text shown on A1:E5 of Figure 14.2. Enter the values in A6:A11. (b) In C2 enter 0.1 for the value of h. Name the cell as h. (c) In B6 enter 0, the initial value of x...
