Modeling and simulation of systems

Slovak University of Technology Faculty of Material Science and Technology in Trnava Modeling and simulation of systems Numerical methods for solving of differential equations

Euler´s method • we solve simple differential equation of first order • y’ = f(x,y), y(x0) = y0 • when we know the value yn= y(xn) solutions at the point xn • and f(xn,yn) = y’(xn) is the directive of tangent to graph of solution at the point (xn,yn) • we can substitute the graph of solution by the line segment with directive f(xn,yn) it means yn+1= y(xn+1) = yn + h f(xn,yn) for small? h on the interval xn,xn +h

Euler´s method y = ex 1 2 3 4 0 Lack of Euler´s method – little precision.

Method of Runge-Kutta • The enlargement of the precision of Euler´s method in the method of Runge-Kutta lies in searching of increasing of at several points on the interval xn,xn+1 • A = (xn,yn) slope at A k1 = h f(xn,yn) • B = (xn+ h/2, yn+ k1/2) slope at B k2 = h f (xn+ h/2, yn+ k1/2) • C = (xn+ h/2, yn+ k2/2) slope at C k3 = h f (xn+ h/2, yn+ k2/2) • D = (xn+ h, yn+ k3) slope at D k4 = h f (xn+ h, yn+ k3)

Method of Runge-Kutta k3 k2 k1 B D k4 C A xn+1 xn xn+2 xn+1+ h/2 xn+ h/2

Method of Runge-Kutta • Method of Runge-Kutta makes the increase – the value yn+1 as a linear combination k1,k2, k3,k4 • Slope of function n = yn+1 - yn n = 1/6 (k1+2k2+2 k3+k4) • Accumulated error is not bigger than constant multiple h5

Predictor-corrector method • For the precise/exact value is in force: It is also necessary to know values of solution in several previous point, this follows from the formula. Let´s substitute integral for closed trapezoid formula: 1

Predictor-corrector method • Let´s subsitute integral for opened trapezoid formula: 2 The first formula is more precise but is in implicit state. First of all we calculate the predictive value yn+1,0 by help of the second formula. Then we set y’n+1,0 and finally we calculate reconstructive value yn+1,1 according to relation 1. This method repeats until the difference of two calculated values is smaller than determined accuracy.

Predictor-corrector method Prediction of valueyn+1,0 Calculationy‘n+1,0= f(xn+1, yn+1,0) 0j Calculation of corrected valueyn+1,j+1 Calculation y‘n+1,j+1= f(xn+1, yn+1,j) j+1  j y‘n+1,j+1- y‘n+1,j N A yn+1= yn+1,j+1y‘n+1 = y‘n+1,j+1

Comparison of methods of numerical integration • The method R-K does not call for additional primary values. There is the possibility to change the step of integration randomly. • The methods P-C call for others primary values. It is generally needed to calculate the primary conditions for the new step at the change of the step of integration. • The accuracy of both methods is approximately the same, often R-K are more precise than P-C of the same rule • R-K needs at each step so much calculation of value f(x,y) as the rule of method. The methods P-C of the fourth rule usually demand two calculation – prediction and correction. That is why we can say that they are about faster twice as R-K of the same order.

The usage of methods of numerical integration • How big is the local defect (error of method, rounding error...) • How influence has the local error upon the results in the next step. Then it is necessary to notify the stability of the method. • The necessity to know the primary conditions • The speed of method

Example Calculate the value(1.5) of solution of differential equationy’=yx, if y(1)=2 and select step 0,1. y0=y(1) a y1=y(1.1) k1= h f(x0,y0)= 0.1*21=0.2

Example k4=h*f(x0+h,y0+k3)=0.1*(2+0.2189)1+0.1=0.2403 y1=y0+1/6*(k1+2k2+2k3+k4)=...=2.219

Modeling and simulation of systems