Mini project ODE, TANA22
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1 Mini project ODE, TANA22 Filip Berglund (filbe882) Linh Nguyen (linng299) Amanda Åkesson (amaak531) October
2 1 Introduction Carl David Tohmé Runge ( ) was a German mathematician and a prominent figure in applied mathematics. Some of Runge s most well-known achievements are the Runge-Kutta methods for numerical solutions of ordinary differential equations which Runge later on, co-developed with another German mathematician, Martin Wilhelm Kutta ( ). The Runge-Kutta methods are used to approximate solutions to ordinary differential equations. The Runge- Kutta methods are a group of implicit and explicit continual methods which are applied to transient problems that occur in the fields of applied physics and engineering. There are quite a number of Runge-Kutta methods, e.g. forward Euler, Heun s method, Lobatto methods, etc. This project mainly focuses on the classical Runge-Kutta method and Fehlberg s method[1]. 2 Assignment For the mini project ODE, we need to construct two MATLAB-functions: myode which solves a first order ordinary differential equation (including system) with initial conditions by using the Classical Runge-kutta method. felhberg which solves a first order ordinary differential equations with initial condition including y-values in Butcher tableau for Fehlberg s 4(5) method (RK45). The report shall also illustrate the correctness of the constructed codes and the order of accuracy by using different step-lengths. 3 Theory The Classical Runge-Kutta method approximates the solution of initial value problem y (x) = f(x, y), y(x 0 ) = y 0. where y n+1 = y n (k 1 + 2k 2 + 2k 3 + k 4 ), k 1 =hf(x n, y n ), k 2 =hf(x n + h 2, y n + k 1 2 ), k 3 =hf(x n + h 2, y n + k 2 2 ), k 4 =hf(x n + h, y n + k 3 ). The local truncation error is O(h 5 ) and the global truncation error is O(h 4 ). 2
3 The Runge-Kutta-Fehlberg method of order 4 has total 5 steps. Each step has specific values in the following order: k 1 =hf (x i, y i ) ( k 2 =hf x i h, y i + 1 ) 4 hk 1 ( k 3 =hf x i h, y i hk ) 32 hk 2 ( k 4 =hf x i h, y i hk hk ) 2197 hk 3 ( k 5 =hf x i + h, y i hk 1 8hk hk ) 4104 hk 4 The approximation to the solution is y i+1 = y i k k k k 5. The global truncation error is O(h 4 ). If Re(λ) < 0 then the system { y = λy, x > 0, y(0) = c + ε has a stable solution and if the error caused by the small perturbation ε in y(0) decreases as x + then the numerical method used to solve the equation is stable. 4 Solution The global truncation error of the Fehlberg method and the Classical Runge- Kutta method on the form O(h p ) will be determined by testing the numerical solution of the differential equation y(x) = y(x) with initial value y(0) = 1 against the exact solution. The exact solution is y(x) = e x and the error will be measured at x = 5 for 10 different step lengths h in the interval [0.0005, 0.5]. Assume that the global truncation error R T (h) = y 5 y(5), where y 5 is the numerical solution, can be approximated by ch p for sufficiently small h, where c is some real number. Taking the logarithm of both sides and letting ȳ = log R T (h) and x = log h yields ȳ = log c + px. If a first degree polynomial is fitted to this linear equation using MATLAB s polyfit, p can easily be identified with the slope of this line. 3
4 The stability of the Classical Runge-Kutta method and the Fehlberg method will be tested on the system { y = 10y, y(0) = 1 + ε with different step lengths h by utilizing the fact that the small perturbation ε will decrease as x + if the method is stable. For this equation we can assume that if the perturbation has decreased at x = 10 it will continue to decrease as x +. 5 Code The MATLAB implementations myode of the Classical Runge-Kutta method and fehlberg of the Fehlberg method. 5.1 myode function y= myode (ffun,x0,xn,y0,h) % Solves the initial value problem y '= ffun (y), y(x0)= y0, % using step length h with the Classical Runge - Kutta method. % Gives the solution at y( xn). y=y0; x=x0; while x < xn -h/2 if nargout ==0 hold on plot (x,y,'.'); hold off k1 = ffun (x,y); k2 = ffun (x+h/2,y+h*k1 /2) ; k3 = ffun (x+h/2,y+h*k2 /2) ; k4 = ffun (x+h,y+h*k3); y=y+(h/6) *( k1 +2* k2 +2* k3+k4); x=x+h; 4
5 5.2 fehlberg function [y, err ] = fehlberg (ffun,x0,xn,y0,h) % Solves the initial value problem y '= ffun (y), y(x0)= y0, % using step length h with Felhbergs method. Gives the % solution at y( xn). Note that err is only an approximation % of the error. A =[0 1/4 3/8 12/ / 2]; B= [ / / 32 9/ / / / / / / / / / /40 0]; Y =[ 25/ / / /5] '; E =[1/ / / /50] '; y=y0; x=x0; while x < xn -h/2 if nargout ==0 hold on plot (x,y,'.'); hold off k1=h* feval (ffun,x,y); k2=h* feval (ffun,x+a (2) *h,y+b (2,1) *k1); k3=h* feval (ffun,x+a (3) *h,y+[ k1 k2 ]*B(3,1:2) '); k4=h* feval (ffun,x+a (4) *h,y+[ k1 k2 k3 ]*B(4,1:3) '); k5=h* feval (ffun,x+a (5) *h,y+[ k1 k2 k3 k4 ]*B(5,1:4) '); y=y+[ k1 k2 k3 k4 k5 ]*Y; x=x+h; err = norm ([ k1 k2 k3 k4 k5 ]*E); 5
6 5.3 Validation Figure 1: Validation of myode and fehlberg on differential equations. initial values are y(0) = 1 in both cases. The Figure 2: Validation of myode and fehlberg on differential systems. The upper system has the initial values y(0) = 1.1, y (0) = 11, and the lower system is originally from Laboration 4[2]. Its solution is made by one of MATLAB s built in functions ode45 instead of the exact one. 6
7 In Figure 1 and 2 we can see that the results of the methods are valid for both differential equations and systems if a well-fitted h is chosen. 6 Results and answers 10 0 Global truncation error for Classical Runge-Kutta Classical Runge-Kutta Fitted curve 10 0 Global truncation error for Fehlberg Fehlberg Fitted curve R T (h) R T (h) h h Figure 3: Global truncation error for y (x) = y(x), y(0) = 1 at x = 5 using different step lengths h, and the curve fitted to those errors using MATLAB s polyfit, as described in the theory section. The order of accuracy is the slope of the fitted curves. In Figure 3 we can see that for fehlberg the slope is thus R T (h) = O(h 4 ) and for myode the slope is thus R T (h) = O(h 4 ) 7
8 7 6 Stability of 10-4 Classical Runge-Kutta Resulting error in the solution at x=10 Perturbation in the initial value Stability of Fehlberg Resulting error in the solution at x=10 Perturbation in the initial value 5 (0.2785, ) 5 (0.302, ) R B (h) 4 3 R B (h) h h Figure 4: The error in y(10) caused by the small perturbation ε when solving y (x) = 10y(x), y(0) = 1 + ε with ε = In Figure 4 we can see that the largest tested step length that yields a stable method for Fehlberg is and for Classical Runge-Kutta is Thus an upper bound for when the respective methods are stable is h and h Discussion The codes work properly and the global truncation error is O(h 4 ) for both methods, agreeing with the theory. On the equation used to test the stability, the Fehlberg method proved stable for bigger step lengths than Classical Runge- Kutta. This might be due to the fact that the Fehlberg method has one more k variable than the Classical Runge-Kutta method. References [1] NE Nationalencyklopedin AB, Carl Runge [2] TANA22, Laboration 4, exercises [3] Nick Trefethen, Stability regions of ODE formulas 8
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