Appendix G: Numerical Solution to ODEs

Transcription

1 Appendix G: Numerical Solution to ODEs The numerical solution to any transient problem begins with the derivation of the governing differential equation, which allows the calculation of the rate of change of the dependent parameter to be computed as a function of the value of the dependent parameter and time. The solution to the problem is therefore a matter of integrating the governing differential equation forward in time. There are a number of techniques available for numerical integration, each with its own characteristic level of accuracy, stability, and complexity. These numerical integration techniques are introduced and discussed in this appendix and revisited throughout the text in order to solve thermonamics problems. In this text we will exclusively use the EES Integral command to numerically solve ordinary differential equations. The EES Integral command automatically implements a sophisticated, adaptive step-size integration technique and is described in Section G.5. Readers interested in only in the implementation details associated with the Integral command in EES should skip directly to Section G.5. Sections G.1 through G.4 provide an overview of the theory behind numerical integration and some of the most common techniques. The governing differential equation that provides the rate of change of a variable (or several variables) given its own value (or values) is sometimes referred to as the state equation for the namic system. State equations characterize the transient behavior of problems in many areas including controls, namics, kinematics, fluids, heat transfer, and electrical circuits. Therefore, the numerical solution techniques provided in this section are relevant to a wide range of engineering problems. In this section, a very simple ODE is solved numerically: 4ty 2t (G-1) with initial condition y = 0 at t = 0. This ODE is relatively simple and therefore an analytical solution can be determined and used as the basis for evaluating the accuracy of the numerical solutions that are derived in this section. Equation (G-1) can be separated: 2 4 y t (G-2) and integrated: y 0 0 t t 2 4y (G-3) The variable w is defined as: w 2 4y (G-4)

2 The differential of w is: dw 4 (G-5) Substituting Eqs. (G-4) and (G-5) into Eq. (G-3) provides: 24y 2 0 t dw t 4 w (G-6) Carrying out the integration in Eq. (G-6) provides: y t ln (G-7) Equation (G-7) is solved for y: The analytical solution is entered in EES: y 1 1 exp 2t (G-8) y = -1/2+exp(-2*t^2)/2 "analytical solution" and used to generate Figure G-1. 0 Dependent variable, y Independent variable, t Figure G-1: Analytical solution. The solution to this problem can also be obtained numerically using one of several available numerical integration techniques. Each numerical technique requires that the total simulation time (t sim ) be broken into small time steps. The simplest possibility uses equal-sized steps, each with duration t:

3 t t sim M 1 (G-9) where M is the number of times at which the solution will be evaluated. The solution at each time step (y j ) is computed by the numerical model, where j indicates the time step (y 1 is the initial condition at t 1 = 0). The time corresponding to each time step is therefore: j 1 tj tsim for j 1.. M M 1 (G-10) G.1 Euler's Method The solution at the end of each time step is computed based on the value at the beginning of the time step and the governing differential equation. The simplest (and generally the least efficient) technique for numerical integration is Euler s method. Euler s method approximates the rate of change within the time step as being constant and equal to its value at the beginning of the time step. Therefore, for any time step j: yj 1 yj t (G-11) yyj, ttj Because the value of the solution at the end of the time step (y j+1 ) can be calculated explicitly using information at the beginning of the time step (y j ), Euler s method is referred to as an explicit numerical technique. The solution at the end of the first time step (y 2 ) is given by: y y t (G-12) 2 1 yy1, t0 where y y 1, t 0 is computed using the governing differential equation. Equation (G-12) is written for every time step, resulting in a set of explicit equations that can be solved sequentially for y at each discrete time. As an example, we can solve Eq. (G-1) with y 1 = 0 at t 1 = 0. The time step duration and array of times for which the solution will be computed is specified using Eqs. (G-9) and (G-10): t_sim=2.5 "time to be simulated" M=21 "number of time steps" Dt=t_sim/(M-1) "time step duration" duplicate j=1,m t[j]=(j-1)*dt "time at each time step" end The initial value of the variable y is specified as the first element in the array y.

4 y[1]=0 "initial value" It is often useful to carry out a single step in a numerical solution before simulating all of the steps. The value of the time rate of change of y at y = y 1 and t = t 1 is determined using Eq. (G-1) and substituted into Eq. (G-12): "take the first step" [1]+4*t[1]*y[1]=-2*t[1] y[2]=y[1]+[1]*dt Once we have established that it is possible to take one step it is easy to take all of the steps. Comment out the code used to take a single step: {"take the first step" [1]+4*t[1]*y[1]=-2*t[1] y[2]=y[1]+[1]*dt} and setup a duplicate loop that goes from j = 1 to j = (M - 1). Copy the code to take one step and place it inside the duplicate loop. Change the element index 1 to j and the element index 2 to j +1. "take all of the steps" duplicate j=1,(m-1) [j]+4*t[j]*y[j]=-2*t[j] y[j+1]=y[j]+[j]*dt end Solving the EES code results in a solution for y at every value of t in the Arrays table. The numerical solution is overlaid onto the analytical solution in Figure G-2. Clearly the numerical solution is approximate. However, the approximation improves as the number of time steps used in the simulation increases (i.e., the duration of the time step is reduced). Figure G-2 shows that numerical simulation more closely approaches the analytical solution if the time step duration is reduced from t = s to t = 0.05 s. Dependent variable, y numerical solution Dt = s numerical solution Dt = 0.05 s analytical solution Independent variable, t Figure G-2: Euler solution with t = s and t = 0.05 s overlaid onto the analytical solution.

5 The analytical solution allows us to compute the error associated with the numerical solution based on the discrepancy between the numerical and analytical solutions at each value of time. The maximum error at any time step, sometimes referred to as the global error, is computed using the Max function. duplicate j=1,m y_an[j] = -1/2+exp(-2*t[j]^2)/2 err[j]=abs(y_an[j]-y[j]) end error=max(err[1..m]) "analytical solution" "error" "maximum or global error" Figure G-3 illustrates the global error associated with Euler's technique as a function of the time step duration Euler's technique Global error Heun's technique Time step Figure G-3: Global error as a function of the time step duration for the Euler technique. Figure G-3 shows that the global error is proportional to the duration of the time step to the first power and therefore Euler's technique is referred to as a first order technique. This characteristic of the solution can be inferred by examination of Eq. (G-12), which is essentially the first two terms of a Taylor series expansion of the function y about time t = 0: d y t d y t y2 y1 t yy1, t0 2! 3! yy1, t0 yy1, t0 Euler's approximation local err neglected terms (G-13) Examination of Eq. (G-13) shows that the local error associated with each time step corresponds to the neglected terms in the Taylor s series and is therefore approximately proportional to t 2 (neglecting the smaller, higher order terms). The local error accumulates during a simulation and becomes the global error. Assuming that the accumulation of local error results in an error that is proportional to the product of the number of time steps and the local error provides:

6 2 global error M t (G-14) The number of time steps is given by: M t sim t (G-15) Substituting Eq. (G-15) into Eq. (G-14) provides: global error t (G-16) which agrees with the error characteristic observed in Figure G-3. Most numerical techniques that are commonly used have higher order and therefore achieve higher accuracy. The other drawback of Euler s technique (and any explicit numerical technique) is that it may become unstable if the duration of the time step becomes too large. For our example, if the time step duration is increased above approximately 0.6 s then the numerical solution becomes unstable, as shown in Figure G Dependent variable, y analytical solution -2 Euler solution with t = s Independent variable, t Figure G-4: Analytical and numerical solution with t = s. G.2 Heun's Method Euler s method is the simplest example of a numerical integration technique; it is an explicit technique with first order accuracy. In this section, Heun s method is presented. Heun's method is an explicit technique with second order accuracy (but with the same stability characteristics as Euler's method). Heun s method is an example of a predictor-corrector technique. In order to simulate any time step j, Heun s method begins with an Euler step to obtain an initial prediction for the solution at

7 the conclusion of the time step ( yˆ j 1 ). This first step in the solution is referred to as the predictor step and the details are essentially identical to Euler s method: yˆ j1 yj t (G-17) yyj, ttj However, Heun s method uses the results of the predictor step to carry out a corrector step. The solution predicted at the end of the time step ( yˆ j 1 ) is used to predict the rate of change at the end of the time step ( ). The corrector step predicts the solution at the end of the time yyˆ j1, ttj1 step (y j+1 ) based on the average of the time rates of change at the beginning and end of the time step. y j1 t yj yy, ˆ 1, 2 j tt j yyj ttj1 (G-18) Heun s method is illustrated in the context of the simple problem discussed previously. The process of moving through the first time step begins with the predictor step: yˆ y t (G-19) 2 1 yy1, tt1 where Eq. (G-1) is used to evaluate y y1, tt1. "take the first step" [1]+4*t[1]*y[1]=-2*t[1] y_hat[2]=y[1]+[1]*dt "time rate of change at t[1]" "predictor step" The corrector step follows: y t y yy1, tt 1 yyˆ 2, tt (G-20) where Eq. (G-1) is also used to evaluate yyˆ 2, tt2. _hat[2]+4*t[2]*y_hat[2]=-2*t[2] y[2]=y[1]+([1]+_hat[2])*dt/2 "time rate of change at t[2] based on y_hat[2]" "corrector step" Once we have established that it is possible to take one step, it is easy to take all of the steps. Comment out the code used to take a single step:

8 {"take the first step" [1]+4*t[1]*y[1]=-2*t[1] y_hat[2]=y[1]+[1]*dt _hat[2]+4*t[2]*y_hat[2]=-2*t[2] y[2]=y[1]+([1]+_hat[2])*dt/2 "time rate of change at t[1]" "predictor step" "time rate of change at t[2] based on y_hat[2]" "corrector step"} and setup a duplicate loop that goes from j = 1 to j = (M - 1). Copy the code to take one step and place it inside the duplicate loop. Change the element index 1 to j and the element index 2 to j +1. "take all of the steps" duplicate j=1,(m-1) [j]+4*t[j]*y[j]=-2*t[j] y_hat[j+1]=y[j]+[j]*dt _hat[j+1]+4*t[j+1]*y_hat[j+1]=-2*t[j+1] y[j+1]=y[j]+([j]+_hat[j+1])*dt/2 end "time rate of change at t[j]" "predictor step" "time rate of change at t[j+1] based on y_hat[j+1]" "corrector step" Figure G-5 illustrates the analytical solution as well as the Euler's solution and Heun's solution with t = s. Notice the improvement in accuracy associated with the use of Heun's technique for the same time step. Figure G-3 illustrates the global error for Heun's technique as a function of the time step and shows more clearly the improvement in accuracy. Notice that the global error is proportional to the time step to the second power; Heun's technique is second order with respect to global error. Dependent variable, y Euler solution Dt = s Heun's solution Dt = s analytical solution Independent variable, t Figure G-5: Analytical and numerical solutions with Euler's and Heun's technique for t = s. Heun s method is a two-step predictor/corrector technique that improves the order of accuracy by one (i.e., the order of Heun s method is two whereas the order of Euler s method is one). It is possible to carry out additional predictor/corrector steps and further improve the accuracy of the numerical solution. One of the most popular higher order techniques is the fourth order Runge- Kutta method (which involves four predictor/corrector steps and is therefore fifth order accurate).

9 G.3 Fully Implicit Method The methods discussed thus far are explicit; they all therefore share the characteristic of becoming unstable when the time step exceeds a critical value. An implicit technique avoids this problem. The fully implicit method is similar to Euler s method in that the time rate of change is assumed to be constant throughout the time step. However, the time rate of change is computed at the end of the time step rather than the beginning. Therefore, for any time step j: y y t (G-21) j1 j yyj1, ttj1 The time rate of change at the end of the time step depends on the solution at the end of the time step (y j+1 ). Therefore, y j+1 cannot be calculated explicitly using information at the beginning of the time step (y j ) and instead an implicit equation is obtained for y j+1. Because EES solves implicit equations, it is easy to implement the fully implicit technique. "Fully implicit technique" duplicate j=1,(m-1) [j+1]+4*t[j+1]*y[j+1]=-2*t[j+1] y[j+1]=y[j]+[j+1]*dt end "time rate of change at t[j+1] and y[j+1]" "fully implicit solution" The fully implicit solution does not become unstable even when the duration of the time step is greater than the critical time step. Figure G-6 illustrates the analytical solution as well as the Euler and fully implicit solution with t = s. Notice that the fully implicit solution remains stable while the Euler's solution does not. Dependent variable, y analytical solution Euler solution with t = s -2 Fully implicit solution with t = s Independent variable, t Figure G-6: Euler's technique and fully implicit technique with t = s.

10 G.4 Crank-Nicolson Method The Crank-Nicolson method is second order and implicit. The time rate of change for the time step is estimated based on the average of its values at the beginning and end of the time step. Therefore, for any time step j: y j1 t yj yy, 1, 2 j tt j yyj ttj1 (G-22) Notice that the Crank-Nicolson is an implicit method because the solution for y j+1 involves a time rate of change that must be evaluated based on y j+1. Therefore, the technique will have the stability characteristics of the fully implicit method. The solution also involves two estimates for the time rate of change and will therefore be second order accurate with respect to the global error. Because EES solves implicit equations, it is relatively easy to implement a Crank- Nicolson solution using EES. "Crank-Nicolson technique" duplicate j=1,m [j]+4*t[j]*y[j]=-2*t[j] end duplicate j=1,(m-1) y[j+1]=y[j]+([j]+[j+1])*dt/2 end "time rate of change at t[j] and y[j]" "Crank-Nicolson solution" The Crank-Nicolson method is a popular choice because it combines high accuracy with stability. G.5 EES' Integral Command The implementation of the techniques discussed thus far has used a fixed duration time step for the entire simulation. This approach is often not efficient because there are regions of time during the simulation where the solution is not changing substantially and therefore large time steps could be taken with little loss of accuracy. Adaptive step-size solutions adjust the size of the time step used based on the local characteristics of the state equation. Typically, the absolute value of the local time rate of change or the second derivative of the time rate of change is used to establish a step-size that is as large as possible but guarantees a certain level of accuracy. For the current example, shown in Figure G-1, smaller time steps would be used near t = 0 s because the solution is changing quickly at this time. Later in the simulation, for t > 2 s, the solution is not changing substantially and therefore large time steps could be used. The implementation of adaptive step size solutions is beyond the scope of this appendix. However, an iterative, second order numerical integration routine that optionally uses an adaptive step-size is provided with EES and can be accessed using the Integral command. EES Integral command requires four arguments and allows an optional fifth argument: F = Integral(Integrand,VarName,LowerLimit,UpperLimit,StepSize)

11 where Integrand is the EES variable or expression that must be integrated, VarName is the integration variable, LowerLimit and UpperLimit define the limits of integration, and StepSize provides the duration of the time step. When using the Integral technique (or indeed any numerical integration technique) it is useful to first verify that, given the integration variable and time, the EES code is capable of computing the time rate of change of the variable. Therefore, the first step is to implement Eq. (G-1) for arbitrary (but reasonable) values of t and y: "Integral Solution" y=0 "arbitrary value of y" t=0 "arbitrary value of t" +4*t*y=-2*t "state equation" which leads to = 0. The next step is to comment out the arbitrary values of y and t that are used to test the computation of the state equation and instead let EES' Integral function control these variables for the numerical integration. The solution is given by: t sim y yini (G-23) 0 where y ini = 0 is the initial condition. Therefore, the solution to our example problem is obtained by calling the Integral function; Integrand is replaced with the variable, VarName with t, LowerLimit with 0, UpperLimit with the variable t_sim, and StepSize with DELTAt, the specified duration of the time step. "Integral Solution" {y=0 "arbitrary value of y" t=0 "arbitrary value of t"} +4*t*y=-2*t "state equation" t_sim=2.5 Dt=0.1 y=integral(,t,0,t_sim,dt) "simulation time" "time step duration" "solution obtained using the integral command" Note that array variables and duplicate loops are not needed to solve an ordinary differential equation when the EES Integral function is used. The elimination of array variables allows much larger problems to be solved and improves the computational speed. In order to accomplish the numerical integration, EES will adjust the value of variable t from 0 to t_sim in increments of Dt. At each value of time, EES will iteratively solve all of the equations in the Equations window that depend on t. For the example above, this process will result in the variable y being evaluated at each value t. When the solution converges, the value of the variable t is incremented and the process is repeated until time is equal to t_sim. The results shown in the Solution window will provide the temperature at the end of the process (i.e., the result of the integration which is y at t = t sim ).

12 Often it is most interesting to know the variation of the solution with time during the process. This information can be provided by including the $IntegralTable directive in the file. The format of the $IntegralTable directive is: $IntegralTable VarName: Step, x,y,z where VarName is the integration variable; the first column in the Integral table will display values of this variable. The colon followed by the parameter Step (which can either be a number or a variable name) and list of variables (x, y, z) are optional. If these values are provided, then the value of Step will be used as the output step size and the integration variables will be reported in the Integral table at the specified output step size. The output step size may be a variable name, rather than a number, provided that the variable has been set to a constant value preceding the $IntegralTable directive. The step size that is used to report integration results is totally independent of the duration of the time step that is used in the numerical integration. If the numerical integration step size and output step size are not the same, then linear interpolation is used to determine the integrated quantities at the specified output steps. If an output step size is not specified, then EES will output all specified variables at every time step. The variables x, y, z... must correspond to variables in the EES program. Algebraic equations involving variables are not accepted within the $Integral directive. A separate column will be created in the Integral table for each specified variable. The variables must be separated by a space or list delimiter (comma or semicolon). Solving the EES code will result in the generation of an Integral table that is filled with intermediate values resulting from the numerical integration. The values in the Integral table can be plotted, printed, saved, and copied in exactly the same manner as for other tables. The Integral table is saved when the EES file is saved and the table is restored when the EES file is loaded. If an Integral table exists when calculations are initiated, it will be deleted if a new Integral table is created. Use the EES code below to generate an Integral Table containing the results of the numerical simulation and the analytical solution. $IntegralTable t, y After running the code with a time step duration of 0.1 s, the Integral table shown in Error! Reference source not found. G-7 will be generated.

13 Figure G-7: Integral Table generated by EES. The results of the integration can be plotted by selecting the Integral table as the source of the data to plot. The StepSize input to the Integral command is optional. If the parameter StepSize is not included or if it is set to a value of 0, then EES will use an adaptive step-size algorithm that maintains accuracy while maximizing computational speed. The parameters used to control the adaptive step-size algorithm can be accessed and adjusted by selecting Preferences from the Options menu and selecting the Integration tab, Figure G-8. The settings can also be provided with a $IntegralAutoStep directive. Figure G-8: Integration preferences dialog. The user can specify how often the step interval will be examined and adjusted as well as the absolute minimum and maximum number of integration steps that will be allowed. The relative error criteria used to either reduce or increase the step size can also be specified. It is advisable to use the $UnitSystem directive to set the unit system so that your code produces consistent results across various computers and is as transparent as possible. For the same reason, it is advisable to use the $IntegralAutoStep directive to set the parameters that govern the numerical integration. For example, the code: $IntegralAutoStep Vary=1 Min=5 Max=2000 Reduce=1e-1 Increase=1e-3

14 specifies these parameters. The integration step size is varied after each step (Vary=1) and the number of steps must be greater than 5 (Min=5) and less than 2000 (Max=2000). The integration step will be reduced if the relative error is larger than 1x10-1 (Reduce=1e-1) and increased if the relative error is less than 1x10-3 (Increase=1e-3). Remove the specified time step from the Integral command and include the $IntegralAutoStep directive: $IntegralAutoStep Vary=1 Min=5 Max=2000 Reduce=1e-1 Increase=1e-3 "Integral Solution" {y=0 "arbitrary value of y" t=0 "arbitrary value of t"} +4*t*y=-2*t "state equation" t_sim=2.5 y=integral(,t,0,t_sim) $IntegralTable t, y "simulation time" "solution obtained using the integral command" "save results in an Integral table" in order to generate the numerical solution shown in Figure G-9. Notice the non-uniform time step duration selected to maintain accuracy while maximizing the computational speed. 0 Dependent variable, y Independent variable, t Figure G-9: Numerical solution obtained with EES' Integral command.