Technology & Engineering
Improved Euler Method
The Improved Euler Method is a numerical technique used to approximate the solutions of ordinary differential equations. It is an enhancement of the standard Euler method and provides more accurate results by using a weighted average of the slope at the beginning and end of each step. This method is widely used in engineering and scientific computing for solving differential equations.
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8 Key excerpts on "Improved Euler Method"
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
This difficulty can be overcome by replacing +1 on the right-hand side of equation (4) by the value obtained using the Euler formula (3). Thus +1 = + ( , ) + ( + ℎ, + ℎ( , )) 2 ℎ = + + ( + ℎ, + ℎ ) 2 ℎ, (5) where +1 has been replaced by + ℎ. 8.2 Improvements on the Euler Method 373 Equation (5) gives an explicit formula for computing +1 , the approximate value of ( +1 ), in terms of the data at . This formula is known as the improved Euler formula or the Heun 1 formula. The improved Euler formula is an example of a two-stage method; that is, we first calculate + ℎ from the Euler formula and then use this result to calculate +1 from equation (5). The improved Euler formula (5) is an improvement over the Euler formula (3) because the local truncation error in using equation (5) is proportional to ℎ 3 , while for the Euler method it is proportional to ℎ 2 . This error estimate for the improved Euler formula is established in Problem 12. It can also be shown that for a finite interval, the global truncation error for the improved Euler formula is bounded by a constant times ℎ 2 , so this method is a second-order method. Note that this greater accuracy is achieved at the expense of more computational work, since it is now necessary to evaluate (, ) twice in order to go from to +1 . If (, ) depends only on and not on , then solving the differential equation ′ = (, ) reduces to integrating (). In this case the improved Euler formula (5) becomes +1 = + ℎ 2 (( ) + ( + ℎ)), (6) which is just the trapezoid rule for numerical integration. EXAMPLE 8.2.1 Use the improved Euler formula (5) with step sizes of ℎ = 0.025 and ℎ = 0.01 to calculate approximate values of the solution of the initial value problem ′ = 1 − + 4, (0) = 1 (7) on the interval 0 ≤ ≤ 2. Solution To make clear exactly what computations are required, we show a couple of steps in detail.
- Barbara D. MacCluer, Paul S. Bourdon, Thomas L. Kriete(Authors)
- 2019(Publication Date)
- American Mathematical Society(Publisher)
Chapter 3 Numerical Methods 3.1. Euler’s method In spite of our efforts in the preceding chapter, the sad fact remains that most first-order initial value problems (3.1) dy dt = f ( t,y ) , y ( t 0 ) = y 0 cannot be solved exactly. In practice, we often have to be content with a graphical or approximate numerical solution, typically produced with the aid of a computer. In this section we discuss Euler’s method, which is one way that a numerical solution is obtained, and investigate the expected error in this method. This is a small part of a big subject, the study of numerical methods in differential equations. Euler’s method is conceptually simple and illustrates some of the basic ideas common to all numerical solution methods. Improvements of Euler’s method are discussed in Section 3.2. In Section 3.3 we will use numerical solution techniques in a number of modeling applications, including an analysis of certain optical illusions and a model for phosphorus accumulation in a lake (see Exercise 7 in Section 3.3). Euler’s method. Euler’s method is based on the same idea used to produce a graphical solution from a direction field. Recall that a direction field for the equation dy dt = f ( t,y ) shows short segments (“field marks”) of slope f ( t,y ) at various points in the ty -plane, as in Fig. 3.1. Euler’s method is a numerical implementation of the following procedure: Pick a starting point ( t 0 ,y 0 ) (i.e., an initial condition), and move a short way from this point in the direction of the field mark at ( t 0 ,y 0 ), stopping at a new point ( t 1 ,y 1 ). We assume that movement along the field marks is in the direction of increasing time, so t 1 > t 0 , and we are moving to the right. From the point ( t 1 ,y 1 ), move in the direction of the field mark at that point for a short distance stopping at a point that we denote ( t 2 ,y 2 ). Continue this process, each time using the field mark at the stopping point to determine in what direction to move next.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
This illustrates that, compared to the Euler method, the Improved Euler Method is clearly more efficient, yielding substantially better results or requiring much less total computing effort, or both. The percentage errors at t = 2 for the Improved Euler Method are 1.23% for h = 0. 025 and 0.21% for h = 0. 01. TABLE 8.2.1 A Comparison of Results Using the Euler and Improved Euler Methods for the Initial Value Problem y = 1 − t + 4 y , y (0) = 1 Euler Improved Euler t h = 0.01 h = 0.001 h = 0.025 h = 0.01 Exact 0. 0 1. 0000000 1. 0000000 1. 0000000 1. 0000000 1. 0000000 0. 1 1. 5952901 1. 6076289 1. 6079462 1. 6088585 1. 6090418 0. 2 2. 4644587 2. 5011159 2. 5020618 2. 5047827 2. 5053299 0. 3 3. 7390345 3. 8207130 3. 8228282 3. 8289146 3. 8301388 0. 4 5. 6137120 5. 7754845 5. 7796888 5. 7917911 5. 7942260 0. 5 8. 3766865 8. 6770692 8. 6849039 8. 7074637 8. 7120041 1. 0 60. 037126 64. 382558 64. 497931 64. 830722 64. 897803 1. 5 426. 40818 473. 55979 474. 83402 478. 51588 479. 25919 2. 0 3029. 3279 3484. 1608 3496. 6702 3532. 8789 3540. 2001 A computer program for the Euler method can be readily modified to implement the Improved Euler Method instead. All that is required is to replace Step 6 in the algorithm in Section 8.1 by the following: The Improved Euler Method Step 6. k 1 = f ( t , y ) k 2 = f ( t + h , y + h ∗ k 1) y = y + ( h / 2) ∗ ( k 1 + k 2) t = t + h 366 CHAPTER 8 Numerical Methods Variation of Step Size. In Section 8.1 we mentioned the possibility of adjusting the step size as a calculation proceeds so as to maintain the local truncation error at a more or less constant level. The goal is to use no more steps than necessary and, at the same time, to keep some control over the accuracy of the approximation. Here, we will describe how this can be done. First, we choose the error tolerance , which is the local truncation error that we are willing to accept. Suppose that after n steps we have reached the point ( t n , y n ).
eBook - PDF- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2021(Publication Date)
- Wiley(Publisher)
This illustrates that, compared to the Euler method, the Improved Euler Method is clearly more efficient, yielding substantially better results or requiring much less total computing effort, or both. The percentage errors at 𝑡 = 2 for the Improved Euler Method are 1.23% for ℎ = 0.025 and 0.21% for ℎ = 0.01. TABLE 8.2.1 A Comparison of Results Using the Euler and Improved Euler Methods for the Initial Value Problem 𝒚 ′ = 𝟏 − 𝒕 + 𝟒𝒚, 𝒚(𝟎) = 𝟏 Euler Improved Euler 𝒕 𝒉 = 𝟎.𝟎𝟏 𝒉 = 𝟎.𝟎𝟎𝟏 𝒉 = 𝟎.𝟎𝟐𝟓 𝒉 = 𝟎.𝟎𝟏 Exact 0.0 1.0000000 1.0000000 1.0000000 1.0000000 1.0000000 0.1 1.5952901 1.6076289 1.6079462 1.6088585 1.6090418 0.2 2.4644587 2.5011159 2.5020618 2.5047827 2.5053299 0.3 3.7390345 3.8207130 3.8228282 3.8289146 3.8301388 0.4 5.6137120 5.7754845 5.7796888 5.7917911 5.7942260 0.5 8.3766865 8.6770692 8.6849039 8.7074637 8.7120041 1.0 60.037126 64.382558 64.497931 64.830722 64.897803 1.5 426.40818 473.55979 474.83402 478.51588 479.25919 2.0 3029.3279 3484.1608 3496.6702 3532.8789 3540.2001 A computer program for the Euler method can be readily modified to implement the Improved Euler Method instead. All that is required is to replace Step 6 in the algorithm in Section 8.1 by the following: The Improved Euler Method Step 6. 𝑘1 = 𝑓(𝑡, 𝑦) 𝑘2 = 𝑓(𝑡 + ℎ, 𝑦 + ℎ ∗ 𝑘1) 𝑦 = 𝑦 + (ℎ ∕ 2) ∗ (𝑘1 + 𝑘2) 𝑡 = 𝑡 + ℎ Variation of Step Size. In Section 8.1 we mentioned the possibility of adjusting the step size as a calculation proceeds so as to maintain the local truncation error at a more or less constant level. The goal is to use no more steps than necessary and, at the same time, to keep some control over the accuracy of the approximation. Here, we will describe how this can be done. First, we choose the error tolerance 𝜖, which is the local truncation error that 8.2 Improvements on the Euler Method 375 we are willing to accept. Suppose that after 𝑛 steps we have reached the point (𝑡 𝑛 , 𝑦 𝑛 ).
- Albert L. Rabenstein(Author)
- 2014(Publication Date)
- Academic Press(Publisher)
Numerical Methods 8.1 THE EULER METHOD In this chapter we shall describe some techniques that can be used to obtain approximate values of the solution of an initial value problem for a first-order differential equation. Such an initial value problem is of the form y=f(*,y y( x o) = y 0 · (8.1) If h is a small positive number, we write x i = x o + K x 2 = x i + h = x o + 2Λ, x 3 = x 2 + h = x 0 + 3h, and so on. In general, x n = x 0 -f nh. We refer to h as the step size to be used in the numerical method. We also use the abbreviations )>i = y{xi)> yi = y(*i . . . j B = y{x n ) for the exact values of the solution at the points x 1? x 2 , ..., x n . If a formula for the solution is not available, then these values are not known. For example, the problem / = x 2 + y y(0) = 1 appears simple, but no explicit formula for its solution is known. 419 420 Numerical Methods In the methods to be investigated in this chapter, an approximation to y u denoted by w l9 is calculated. Then an approximate value vv 2 for y 2 is found, and so on. These approximate values are calculated step by step, with the value of w 2 depending on w l (and perhaps on w 0 also), while w 3 depends on w 2 (and possibly on w x and w 0 also). If w n depends only on w„_ l (and not on w„_ 2 or other previously calculated values) the method is called a single-step method (as opposed to a multi-step method). We shall investigate several single-step methods and then conclude the chapter with a discussion of a multi-step technique. All the single-step methods we shall discuss are based on Taylor's formula, which is derived in most calculus textbooks. This formula states that (under appropriate conditions 1 ) y(x + h) = y(x) + hyx) + ~ y(x) + ~'+^ / ° Μ + **(*, h where with ξ between x and x + h. In particular, if k = 1, we have y (x + h) = y(x) + hy'(x) + R^x, h and R l is proportional to h 2 .
eBook - PDFDifferential Equations
An Introduction to Modern Methods and Applications
- James R. Brannan, William E. Boyce(Authors)
- 2015(Publication Date)
- Wiley(Publisher)
This illustrates that, compared to the Euler method, 8.3 Improved Euler and Runge–Kutta Methods 541 the Improved Euler Method is clearly more efficient, yielding substantially better results or requiring much less total computing effort, or both. The percentage errors at t = 2 for the Improved Euler Method are 1.23% for h = 0.025 and 0.21% for h = 0.01. ▶ Variation of Step Size. In Section 8.2, we mentioned the possibility of adjusting the step size as a calculation proceeds so as to maintain the local truncation error at a more or less con- stant level. The goal is to use no more steps than necessary and, at the same time, to keep some control over the accuracy of the approximation. Here, we will describe how this can be done. Suppose that after n steps we have reached the point (t n , y n ). We choose a step size h and calculate y n+1 . Next we need to estimate the error we have made in calculating y n+1 . Not knowing the actual solution, the best that we can do is to use a more accurate method and repeat the calculation starting from (t n , y n ). For example, if we used the Euler method for the original calculation, we might repeat it with the Improved Euler Method. Then the difference between the two calculated values is an estimate e est n+1 of the error in using the original method. If the estimated error is different from the error tolerance , then we adjust the step size and repeat the calculation. To make this adjustment efficiently, it is crucial to know how the local truncation error e n+1 depends on the step size h. For the Euler method, the local truncation error is proportional to h 2 , so to bring the estimated er- ror down (or up) to the tolerance level , we must multiply the original step size by the factor √ ∕e est n+1 . To illustrate this procedure, consider the example problem (11): y ′ = 1 − t + 4y, y(0) = 1.
- William E. Boyce, Richard C. DiPrima, Douglas B. Meade(Authors)
- 2017(Publication Date)
- Wiley(Publisher)
To illustrate this procedure, consider the example problem (7): y ′ = 1 − t + 4y, y(0) = 1. Suppose that we choose the error tolerance to be 0.05. You can verify that after one step with h = 0.1, we obtain the values 1.5 and 1.595 from the Euler method and the Improved Euler Method, respectively. Thus the estimated error in using the Euler method is 0.095. Since this is larger than the tolerance level of 0.05, we need to adjust the step size downward by the factor √ 0.05∕0.095 ≅ 0.73. Rounding downward to be conservative, let us choose the adjusted step size h = 0.07. Then, from the Euler formula, we obtain y 1 = 1 + (0.07) f (0, 1) = 1.35 ≅ (0.07). Then, using the Improved Euler Method with h = 0.07, we obtain y 1 = 1.39655, so the estimated error in using the Euler formula is 0.04655, which is slightly less than the specified tolerance. The actual error, based on a comparison with the exact solution, is somewhat greater, namely, 0.05122. We can follow the same procedure at each step of the calculation, thereby keeping the local truncation error approximately constant throughout the entire numerical process. Modern adaptive codes for solving differential equations adjust the step size in very much this way as they proceed, although they usually use more accurate formulas than the Euler and improved Euler formulas. Consequently, they are able to achieve both efficiency and accuracy by using very small steps only where they are really needed. Problems N 1. Complete the calculations leading to the entries in columns four and five of Table 8.2.1. In each of Problems 2 through 6, find approximate values of the solution of the given initial value problem at t = 0.1, 0.2, 0.3, and 0.4. Compare the results with those obtained by the Euler method and the backward Euler method in Section 8.1 and with the exact solution (if available). N a. Use the Improved Euler Method with h = 0.05. N b.
eBook - ePubDifferential Equations
A Modern Approach with Wavelets
- Steven Krantz(Author)
- 2020(Publication Date)
- Chapman and Hall/CRC(Publisher)
0 ). Then (5.4.1) becomesy 1=y 0+h 2⋅[.f](+ f)x 0,y 0()x 1,z 1The reader should pause to verify that each quantity on the right-hand side can be calculated from information that we have—without knowledge of the exact solution of the differential equation. More generally, our iterative scheme iswherey=j + 1y j+h 2⋅[,f](+ f)x j,y j()x,j + 1zj + 1z=j + 1y j+ h ⋅ f()x j,y jand j = 0,1, 2,….This new method, usually called the Improved Euler Method or Heun’s method, first predicts and then corrects an estimate for yj . It is an example of a class of numerical techniques called predictor-corrector methods. It is possible, using subtle Taylor series arguments, to show that the local discretization error isFIGURE 5.3The Improved Euler Method.ϵ j= − y ″ ′( ξ )⋅,h 312for some value of ξ between x0 and xn . Thus, in particular, the total discretization error is proportional to h2 (instead of h, as before), so we expect more accuracy for the same step size. Figure 5.3 gives a way to visualize the Improved Euler Method. First, the point at (x1 , z1 ) is predicted using the original Euler method, then this point is used to estimate the slope of the solution curve at x1 . This result is then averaged with the original slope estimate at (x0 , y0 ) to make a better prediction of the solution—namely, (x1 , y1 ).We shall continue to examine our old friendy ′= x + y ,y( 0 )= 1and use the value y(1) as a benchmark.EXAMPLE 5.4.1 Apply the Improved Euler Method to the differential equation
with step size 0.2 and gauge the improvement in accuracy over the ordinary Euler method used in Examples 5.2.1 and 5.3.2. ■y ′= x + y ,y( 0 )= 1(5.4.1.1) TABLE 5.4.1.2. Tabulated values for exact and numerical solutions to (5.4.1.1) with h
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