TP 1: Euler s Algorithm-Air Resistance-Introduction to Fortran

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1 TP 1: Euler s Algorithm-Air Resistance-Introduction to Fortran December 10, References N.J.Giordano, Computational Physics. R.H.Landau, M.J.Paez, C.C.Bordeianu, Computational Physics. H.Gould, J.Tobochnick, D.Christian, An Introduction To Computer Simulation Methods. R.Fitzpatrick, Computational Physics. 2 Introduction Computational physics is a subfield of computational science/scientific computing. In computational physics we combine elements from physics (especially theoretical), elements from mathematics (in particular applied mathematics such as numerical analysis) and elements from computer science (programming) for the purpose of solving a physics problem. In physics there are traditionally two approaches. 1) The experimental approach and 2) The theoretical approach. Nowadays we also consider as a third approach in physics namely: The computational approach. It can even be argued that the computational approach is independent from the first two approaches and it is not just a bridge between them. The most important use of computers in physics is simulation. Simulations are suited for nonlinear problems which can not generally solved by analytical methods. The starting point of a simulation is an idealized model of a physical system of interest. We want to check whether or not the behaviour of this model is consistent with observation. We specify an algorithm for the implementation of the model on a computer. The execution of this implementation is a simulation. Simulations are therefore virtual experiments. The comparison between simulations and laboratory experiments goes as follows Laboratory experiment sample physical apparatus calibration measurement data analysis Simulation model computer program (the code) testing of code computation data analysis A crucial tool in computational physics is programming languages. In simulations as used by the majority of research physicists codes are written in a high-level compiled language such as Fortran and C/C++. In such simulations we also use calls to routine libraries such as LAPACK. In this course we follow a different path by writing all our codes in a high-level compiled language and not call any libraries. 1

2 The use of mathematical software packages such as MAPLE, MATHEMATICA and MATLAB is only suited for relatively small calculations. These packages are interpreted languages and thus the code they produce run generally far too slowly compared to compiled languages. As our programming language we will use Fortran 77 under the Linux operating system. We adopt the ubuntu distribution of Linux. We will use the Fortran compilers f77 and gfortran. As an editor we will use emacs and for graphics we will use gnuplot. 3 The Physics-Air Resistance We consider an athlete riding a bicycle moving on a flat terrain. The goal is to determine the velocity. Newton s second law is given by m dv = F. (1) F is the force exerted by the athlete on the bicycle. It is clearly very difficult to write down a precise expression for F. Formulating the problem in terms of the power generated by the athlete will avoid the use of an explicit formula for F. Multiplying the above equation by v we obtain de = P. (2) E is the kinetic energy E = 1 2 mv2 and P is the power P = Fv. Experimentaly we find that the output of well trained athletes is around P = 400 watts over periods of 1h. The above equation can also be rewritten as For P constant we get the solution dv 2 = 2P m. (3) v 2 = 2P m t + v2 0. (4) We remark the unphysical effect that v as t. This is due to the absence of the effect of friction and in particular air resistance. The most important form of friction is air resistance. The force due to air resistance (the drag force) is F drag = B 1 v B 2 v 2. (5) At small velocities the first term dominates whereas at large velocities it is the second term that dominates. For very small velocities the dependence on v given by F drag = B 1 v is known as Stockes law. For reasonable velocities the drag force is dominated by the second term, i.e it is given by F drag = B 2 v 2 for most objects. The coefficient B 2 can be calculated as follows. As the bicycle-rider combination moves with velocity v it pushes in a time a mass of air given by dm air = ρav where ρ is the air density and A is the frontal cross section. The corresponding kinetic energy is de air = dm air v 2 /2. This is equal to the work done by the drag force,i.e F drag v = de air. From this we get The drag coefficient is C = 1 2. The drag force becomes B 2 = CρA. (6) F drag = CρAv 2. (7) Taking into account the force due to air resistance we find that Newton s law becomes Equivalently m dv = F + F drag. (8) dv = P mv CρAv2 m. (9) 2

3 4 Another Example-Radioactive Decay In a spontaneous radioactive decay a particle with no external influence will decay into other particles. A typical example is the nuclear isotope uranium 235. The exact moment of decay of any one particle is random. This means that the number dn(t) = N(t) N(t + ) of nuclei which will decay during a time inetrval must be proportional to and to the number N(t) of particles present at time t. In other words the probability of decay per unit time given by ( dn(t)/n(t))/ is a constant which we denote 1/. (The minus sign is due to the fact that dn is negative since the number of particles decreases with time). We write dn(t) The solution of this first order differential equation is = N(t). (10) N(t) = N 0 exp( t/). (11) The number N 0 is the number of particles at time t = 0. The time is called the mean lifetime. It is the average time for decay. For the uranium 235 the mean lifetime is around 10 9 years. 5 Numerical Analysis-Euler s Algorithm The goal is to obtain an approximate numerical solution to the above problem which in this particular case can be compared to an exact solution given by the exponential decay law (11). We start from Taylor s expansion We get in the limit t 0 N(t + t) = N(t) + t dn ( t)2 d2 N +... (12) 2 dn = Lim N(t + t) N(t) t 0. (13) t We take t small but non zero. In this case we obtain the approximation Equivalently By using (10) we get dn + t) N(t) N(t. (14) t N(t + t) N(t) + t dn. (15) N(t + t) N(t) t N(t). (16) We will start from the number of particles at time t = 0 given by N(0) = N 0 N(1). We substitute t = 0 in (16) to obtain N( t) = N(2) as a function of N(1). Next the value N(2) can be used in equation (16) to get N(2 t) = N(3), etc. We are led to the time discretization t t(i) = (i 1) t, i = 1,..., L + 1. (17) The integer L determine the total time interval T = L t. The numerical solution (16) can be rewritten as (with N(t) = N(i 1)) This is Euler s algorithm. N(i 1) N(i) = N(i 1) t, i = 2,..., L + 1. (18) 3

4 6 Fortran Code The program, return and end statements: Any fortran program will begin with a program statement and will conclude with an end statement. The program statement allows us to give a name to the program. The end statement may be preceded by a return statement. This looks like program radioactivity c here is the code return end (19) We have chosen the name radioactivity for our program. The c in the second line indicates that the sentence here is the code is only a comment and not a part of the code. Variables,Declaration and Types: After the program statement come the declaration statements. We state the variables and their types which are used in the program. In Fortran we have the integer type for integer variables and the double precision type for real variables. In our case the variables N(i), t(i),, t, N 0 are real numbers while the variables i (which is the index of N(i) and t(i)) and L are integer numbers. Array: An array A of dimension K is an ordered list of K variables of a given type called the elements of the array and denoted A(1), A(2),...,A(K). In our example N(i) and t(i) are real arrays of dimension L + 1. In Fortran the default lower limit of the range of an array is 1 and not 0. In Fortran we declare that N(i) and t(i) are real for all i = 1,..., L + 1 by writing N(1 : L + 1) and t(1 : L + 1). The parameter statement: Since an array is declared at the begining of the program it must have a fixed size. In other words the upper limit must be a constant and not a variable. In Fortran a constant is declared in parameter statement. In our case the upper limit is L+1 and hence L must be declared in parameter statement. By putting all declarations together we get program radioactivity integer i, L parameter (L = 100) doubleprecision N(1 : L + 1), t(1 : L + 1), N 0, t, c here is the code return end (20) Input: The input parameters of the problems are N 0, and t. The do loop: For the radioactivity problem the main part of the corresponding code consists of equation (18). We start with the known quantities N(1) = N 0 at t(1) = 0 and generate via the successive use of (18) and (17) N(i) and t(i) for all i > 1. This will be coded using a do loop. It begins with a do statement and ends with an enddo statement. We may also indicate a step size. 4

5 The write statement: The output of the computation can be saved to a file using a write statement. In our case the output is the number of particles N(i) and the time t(i). Then the write statement will read write(10, )t(i), N(i). The data will be saved to a file called fort.10. The do loop with a write statement is done as follows N(1) = N 0 t(1) = 0 do i = 2, L + 1, 1 N(i) = N(i 1) ( t N(i 1))/ t(i) = (i 1) t write(10, )t(i), N(i) enddo (21) 5