Computational Mechanics: Coursework on ODEs SPRING 2013

Transcription

1 Computational Mechanics: Coursework on ODEs SPRING 3 Hand-in date: Thursday st February ( pm) This coursework contributes to your degree and must be your own individual work. You should not confer with others. You are also solely responsible for putting your own coursework in the correct submission box by the date and time above. No marks will be given if working is not shown. 3. The Computer Program The program can be used to solve, numerically, any initial-value problem of the form: dy = F, Y ( x ) = Y In general, Y and F are vectors and F is a function of both x and Y. Program Files The following files can be downloaded from Blackboard or the web page for this module. solve.f95 the Fortran source file solve.in a sample input file Breakdown of Program File solve.f95 PROGRAM SOLVE SUBROUTINE STEP SUBROUTINE DERIV SUBROUTINE OUTPUT FUNCTION EXACT main control program performs one forward step returns the derivative F at given (x,y) performs any required output at given (x,y) optionally, can return the exact solution (if known) Input File solve.in NDEP METHOD NSTEP H NPRINT X Y()... Y(NDEP) number of dependent variables; i.e. dimension of Y method of solution one of EULER, MODIF, RUNGE number of steps step size output frequency initial x initial Y (one value for each component) The basic procedures are already coded for you in subprograms SOLVE and STEP. For any given equation the user should amend the code to specify: derivative F in subroutine DERIV; any required output in subroutine OUTPUT; (optionally) the solution in function EXACT; this is only for testing purposes; numerical parameters and initial values in file solve.in. Computational Mechanics 3 - David Apsley

2 Sample Equations sample equations have been provided in the code (see subprograms DERIV and EXACT). (A) dy x = x + y, y ( ) = (solution: y = e x ) (B) d y dy x y = x, y ( ) =, y () = (solution: x 3 y = 5e e + x ) In the latter case, the system has been reduced to first order with dy Y = y, Y = d Y Y Y =, = at x = Y x Y 3Y Y The user may switch between these sample problems by: un-commenting the appropriate lines in subprograms DERIV and EXACT; setting the number of variables (NDEP) and initial values (X and Y) in solve.in The default output (see subroutine OUTPUT) is both to screen and to a file (solve.out) and consists of the following columns: x y y(exact) error Sample Files solve.in Problem A: solved by the Euler method; step size.; steps; output every step. number of dependent variables (NDEP) EULER method of solution (METHOD) number of steps (NSTEP). step size (H) output frequency (NPRINT). initial x.. initial Y(); number of values corresponds to NDEP Problem B: solved by the Runge-Kutta method; step size.5; steps; output every steps. number of dependent variables (NDEP) RUNGE method of solution (METHOD) number of steps (NSTEP).5 step size (H) output frequency (NPRINT). initial x.. initial Y(); number of values corresponds to NDEP **** Very important **** The number of items of initial data for Y must correspond to the number of dependent variables, NDEP. Computational Mechanics 3 - David Apsley

3 3. Coursework PART A: NUMERICAL METHODS A. Compare Solution Methods For the first equation (problem A), run Euler ('EULER'), modified Euler ('MODIF') and standard Runge-Kutta ('RUNGE') methods to derive the solution in x with step size x =.. Compare all numerical solutions and the exact solution in a single table (using the same precision as the output of the code) and on a single graph. include a table containing y values for all of these methods against x; include the graph requested above; state the maximum absolute error for each method; comment briefly on the relative errors produced by the different methods; state the advantages and disadvantages of using a high-order method like Runge- Kutta compared with a low-order method like Euler. A. Effect of Step Size Using the modified Euler method ('MODIF'), integrate problem A between x = and x = using step sizes x =.,.5,.,. in turn. (You will need to adjust NSTEP as the step size changes, in order that x max =.) Tabulate E, log E and log x against x, where E is the absolute error at x =. Use the same precision as the output of the code. Plot a graph of log E against log x. include a table of absolute error (at x = ) against stepsize; include the graph; state what is meant by an order- numerical scheme and determine whether your graph is consistent with this. Computational Mechanics 3-3 David Apsley

4 PART B: PRACTICAL EXAMPLE: LAMINAR BOUNDARY-LAYER FLOWS x u y The wall-parallel (x) component of velocity in a high-reynolds-number self-similar boundary-layer flow is given by y u = U f (η), where η = () δ Here, U (x) is the free-stream velocity, δ(x) is proportional to boundary-layer depth and f is the normalised stream function. f (η) satisfies the Falkner-Skan equation f + ff + β( f ) = () subject to boundary conditions f ( ) = f () =, f ( ) = (3) β is a parameter that is related to changes in the free stream: β < for decelerating flow, β > for accelerating flow and β = for constant free-stream velocity (zero pressure gradient). B. Governing Equation Show how Equation () can be written as a first-order differential equation with vector dependent variable. explain how this is done (e.g. by stating the components of the relevant vectors). B. Implementation Modify subroutine DERIV to solve Equation (). Modify subroutine OUTPUT to output f, df/dη and d f/dη but no exact solutions or errors (since these are unknown). Leave β as a variable to be modified when necessary. include the modified subroutines DERIV and OUTPUT. For reference (i.e. you don t need to know this to do the question!), if the free-stream velocity U x m then νx δ ( x) =, m β = + m U + m and the case m = gives the famous Blasius boundary-layer profile. Computational Mechanics 3-4 David Apsley

5 B3. Test Cases Compute the solution for three cases: β = : constant free-stream velocity; β =.9: decelerating flow (e.g. near separation); β = : accelerating flow (e.g. away from a stagnation point). In each case employ a shooting method with a sequence of guesses for f (), so as to find one which gives f ( ) =. ( is sufficiently close to in this problem!). The step size and choice of integration method are entirely yours, but will need to be justified. Note that, for non-zero β, solutions can be very sensitive to f () ; you need only search in the range (,.) here. plot a single graph (comparing the results for the three values of ) showing the U normalised velocity, = f (η) against for 5; U state the value of f () used for each value of ; justify the choice of step size and integration method which you have used; include the input file SOLVE.IN for any one of the three values of (but state which!) Computational Mechanics 3-5 David Apsley

6 PART C: ADDITIONAL INTEGRATION METHODS Modify subroutine STEP to allow two additional cases: 'RKFEH' for the Runge-Kutta-Fehlberg method defined by the following tableau 'IMPLI' for the implicit modified Euler method defined by Y Y + x[ F( x, Y ) F( x, Y )] i+ = i i i + i+ i+ You are advised to check that both sample equations can be solved with these methods. Use the Runge-Kutta-Fehlberg method and both explicit and implicit modified Euler methods to solve the equation dy 3 = x 5y, y() = using steps of length x =.. Compare your numerical solutions in a single table and on a graph. include the modified subroutine STEP and, for the equation above, the modified subroutine DERIV; include the table and graph requested above. Computational Mechanics 3-6 David Apsley