Introduction to Finite Di erence Methods

Transcription

1 Introduction to Finite Di erence Methods ME 448/548 Notes Gerald Recktenwald Portland State University Department of Mechanical Engineering ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation Overview 1. Review of basic numerical analysis 2. Finite di erence notation 3. Finite di erence approximations ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 1

2 Preliminary Considerations from Numerical Analysis ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 2 Important Ideas from Numerical Analysis 1. Symbolic versus numeric calculation 2. Numerical arithmetic and the floating point number line 3. Catastrophic cancellation errors 4. Machine precision and roundo 5. Truncation error Note: Examples in these slides will use Matlab. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 3

3 Symbolic versus Numeric Calculation Commercial software for symbolic computation Derive TM MACSYMA TM Maple TM Mathematica TM Symbolic calculations are exact. No rounding occurs because symbols and algebraic relationships are manipulated without storing numerical values. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 4 Symbolic versus Numeric Calculation Example: Evaluate f( ) =1 sin 2 cos 2 with Matlab >> theta = 30*pi/180; >> f = 1 - sin(theta)^2 - cos(theta)^2 f = e-16 f is close to, but not exactly equal to zero because of roundo. Also note that f is a single value, not a formula. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 5

4 Numerical Arithmetic Engineering analysis is usually done with real numbers. Infinite in range. Infinite in precision Computer architecture imposes limits on numerical values. Finite range. Finite precision Finite precision is significantly more problematic than finite range. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 6 Integer Arithmetic Operation Result 2+2=4 integer 9 7=63 integer 12 =4 3 integer 29 =2 13 exact result is not an integer 29 = exact result is not an integer ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 7

5 Floating Point Arithmetic Operation Floating Point Value is =4 exact =63 exact 12.0 =4 exact = approximate = approximate ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 8 Floating Point Number Line Compare floating point numbers to real numbers. Range Precision Real numbers Infinite: arbitrarily large and arbitrarily small real numbers exist. Infinite: Thereisaninfiniteset of real numbers between any two real numbers. Floating point numbers Finite: the number of bits allocated to the exponent limit the magnitude of floating point values. Finite: thereisafinitenumber (perhaps zero) of floating point values between any two floating point values. In other words: The floating point number line is a subset of the real number line. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 9

6 Floating Point Number Line denormal overflow usable range underflow underflow usable range overflow realmax realmin realmin realmax zoom-in view ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 10 Catastrophic Cancellation Errors (1) The errors in c = a + b and c = a b will be large when a b or a b. Consider c = a + b with a = x.xxx b = y.yyy where x and y are decimal digits. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 11

7 Catastrophic Cancellation Errors (1) Evaluate c = a + b with a = x.xxx and b = y.yyy Assume for convenience of exposition that z = x + y < 10. z available precision } { x.xxx xxxx xxxx xxxx yyyy yyyy yyyy yyyy = x.xxx xxxx zzzz zzzz yyyy yyyy {z } lost digits The most significant digits of a are retained, but the least significant digits of b are lost because of the mismatch in magnitude of a and b. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 12 Catastrophic Cancellation Errors (2) For subtraction: Theerrorin will be large when a b. c = a b Consider c = a b with a = x.xxxxxxxxxxx1ssssss b = x.xxxxxxxxxxx0tttttt where x, y, s and t are decimal digits. The digits sss... and ttt... are lost when a and b are stored in double-precision, floating point format. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 13

8 Catastrophic Cancellation Errors (3) Evaluate a b in double precision floating point arithmetic when a = x.xxx xxxx xxxx 1 and b = x.xxx xxxx xxxx 0 z available precision } { x.xxx xxxx xxxx 1 x.xxx xxxx xxxx 0 = uuuu uuuu uuuu {z } unassigned digits = 1.uuuu uuuu uuuu The result has only one significant digit. Values for the uuuu digits are not necessarily zero. The absolute error in the result is small compared to either a or b. Therelative error in the result is large because ssssss tttttt 6= uuuuuu (except by chance). ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 14 Catastrophic Cancellation Errors (4) Summary: Cancellation errors Occur in addition + or subtraction when or Occur in subtraction: when Are caused by a single operation (hence the term catastrophic ) not a slow accumulation of errors. Can often be minimized by algebraic rearrangement of the troublesome formula. (Cf. improved quadratic formula.) Note: roundo errors that are not catastrophic can also cause problems. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 15

9 Machine Precision The magnitude of roundo errors is quantified by machine precision " m. There is a number, " m > 0, suchthat whenever < " m. 1+ =1 In exact arithmetic 1+ =1only when =0,soinexactarithmetic" m is identically zero. Matlab uses double precision (64 bit) arithmetic. The built-in variable eps stores the value of " m. eps = ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 16 Implications for Routine Calculations Floating point comparisons should test for close enough instead of exact equality. Express close in terms of or absolute di erence, x y x y relative di erence, x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 17

10 Floating Point Comparison Don t ask, is x equal to y?. if x==y % Don t do this... end Instead ask, are x and y close enough in value? if abs(x-y) < tol... end ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 18 Practical Implications: Absolute and Relative Error (1) Close enough can be measured with absolute di erence or relative di erence, or both. Let = an exact or reference value, and b = the value we have computed Absolute error Relative error E abs (b ) = b E rel (b ) = b ref Often we choose ref = so that E rel (b ) = b ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 19

11 Absolute and Relative Error (2) Example: Approximating sin(x) for small x Since sin(x) =x x 3 3! + x5 5!... we can approximate sin(x) with for small enough x < 1 sin(x) x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 20 Absolute and Relative Error (3) The absolute error in approximating sin(x) x for small x is E abs = x sin(x) = x3 3! x 5 5! +... And the relative error is E abs = x sin(x) sin(x) = x sin(x) 1 ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 21

12 Absolute and Relative Error (4) Plot relative and absolute error in approximating sin(x) with x. Although the absolute error is relatively flat around x =0,the relative error grows more quickly. 20 x Error in approximating sin(x) with x Absolute Error Relative Error The relative error grows quickly because the absolute value of sin(x) is small near x =0. Error x (radians) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 22 Practical Implications: Iteration termination (1) An iteration generates a sequence of scalar values x k, k =1, 2, 3,... The sequence converges to a limit,, if x k <, for all k>n, where is a small. In practice, the test is usually expressed as x k+1 x k <, when k>n. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 23

13 Absolute convergence criterion Iteration termination (2) Iterate until x x old < a where a is the absolute convergence tolerance. In Matlab: x =... % initialize xold =... while abs(x-xold) > deltaa xold = x; update x end ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 24 Relative convergence criterion Iteration termination (3) Iterate until x x old x ref < r where r is the absolute convergence tolerance. In Matlab: x =... % initialize xold =... xref =... while abs((x-xold)/xref) > deltar xold = x; update x end ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 25

14 Truncation Error: Distinct from Roundo Error Consider the series for sin(x) sin(x) =x x 3 3! + x5 5! For small x, only a few terms are needed to get a good approximation to sin(x). The terms are truncated f true = f sum + truncation error The size of the truncation error depends on x and the number of terms included in f sum. For typical work we can only estimate the truncation error. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 26 Truncation error in the series approximation of sin(x) (1) function ssum = sinser(x,tol,n) % sinser Evaluate the series representation of the sine function % % Input: x = argument of the sine function, i.e., compute sin(x) % tol = tolerance on accumulated sum. % n = maximum number of terms. % % Output: ssum = value of series sum after nterms or tolerance is met term = x; ssum = term; % Initialize series fprintf( Series approximation to sin(%f)\n\n k term ssum\n,x); fprintf( %3d %11.3e %12.8f\n,1,term,ssum); for k=3:2:(2*n-1) term = -term * x*x/(k*(k-1)); % Next term in the series ssum = ssum + term; fprintf( %3d %11.3e %12.8f\n,k,term,ssum); if abs(term/ssum)<tol, break; end % True at convergence end fprintf( \ntruncation error after %d terms is %g\n\n,(k+1)/2,abs(ssum-sin(x))); ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 27

15 Truncation error in the series approximation of sin(x) (2) For small x, theseriesforsin(x) converges in a few terms >> s = sinser(pi/6,5e-9,15); Series approximation to sin( ) k term ssum e e e e e e Truncation error after 6 terms is e-014 ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 28 Truncation error in the series approximation of sin(x) (3) The truncation error in the series is small relative to the true value of sin( /6) >> s = sinser(pi/6,5e-9,15);. >> err = (s-sin(pi/6))/sin(pi/6) err = e-014 ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 29

16 Truncation error in the series approximation of sin(x) (4) For larger x, the series for sin(x) converges more slowly >> s = sinser(15*pi/6,5e-9,15); Series approximation to sin( ) k term ssum e e e e e e Truncation error after 15 terms is e-007 More terms are needed to reach a converge tolerance of ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 30 Taylor Series: a tool for analyzing truncation error For a su ciently continuous function f(x) defined on the interval x 2 [a, b] we define the n th order Taylor Series approximation P n (x) P n (x) =f(x 0 )+(x x 0 ) df + (x x 0) 2 dx x=x0 2 d 2 f + + (x dx 2 x=x 0 n! x 0) n d n f dx n x=x 0 Then there exists with x 0 apple apple x such that f(x) =P n (x) +R n (x) where R n (x) = (x x 0) (n+1) (n + 1)! d (n+1) f dx (n+1) x= ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 31

17 Big O notation Big O notation f(x) =P n (x) +O (x x! 0) (n+1) (n + 1)! or, for x x 0 = h we say f(x) =P n (x) +O h (n+1) The Big O lumps multiplicative factors (like 1/(n + 1)!) into constants which are (usually) ignored. When working with Big O notation, we focus on the power of the term inside the parenthesis. Since h! 0 as we refine the computations, larger powers are more beneficial. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 32 Taylor Series Example Consider the function f(x) = 1 1 x The Taylor Series approximations to f(x) of order 1, 2 and 3 are P 1 (x) = P 2 (x) = P 3 (x) = 1 1 x 0 + x x 0 (1 x 0 ) x x 0 1 x 0 (1 x 0 ) + (x x 0) 2 2 (1 x 0 ) x x 0 1 x 0 (1 x 0 ) + (x x 0) 2 2 (1 x 0 ) + (x x 0) 3 3 (1 x 0 ) 4 ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 33

18 Taylor Series: P i (x) near x =1.6 (5) Approximations to f(x) = 1/(1 x) exact P 1 (x) P 2 (x) P 3 (x) x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 34 Roundo and Truncation Errors (1) Both roundo and truncation errors occur in numerical computation. Example: Finite di erence approximation to f 0 (x) =df /dx f 0 f(x + h) (x) = f(x) h h 2 f 00 (x) +... This approximation is said to be first order because the leading term in the truncation error is linear in h. Dropping the truncation error terms we obtain f 0 fd + h) (x) =f(x f(x) h or f 0 fd (x) =f 0 (x) +O(h) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 35

19 Roundo and Truncation Errors (2) To study the roles of roundo and truncation errors, compute the finite di erence 1 approximation to f 0 (x) when f(x) =e x. The relative error in the f 0 d fd (x) approximation to dx ex is E rel = f 0 fd (x) f 0 (x) f 0 (x) = f 0 fd (x) ex e x 1 The finite di erence approximation is used to obtain numerical solutions to ordinary and partial di erentials equations where f(x) and hence f 0 (x) is unknown. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 36 Roundo and Truncation Errors (3) Evaluate E rel at x =1for a range of h. Truncation error dominates at large h Roundo error in f(x + h) f(h) dominates as h! 0. Relative error Roundoff error dominates Truncation error dominates Stepsize, h ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 37

20 Finite Di erence Approximations to the Heat Equation ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 38 Model Problem: 1D 0 apple x apple L (1) u(0,t)=0, u(l, t) =0 (2) u(x, 0) = u 0 (x) (3) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 39

21 Example: Place a hot pot on a table. What is the transient temperature distribution in the table? t < 0 t > 0 L x 0 x T L T T 0 increasing time L ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 40 Model Problem: 1D Heat Equation Exact Solution u(x, t) = 1X 1X u n (x, t) = A n exp( n=1 n=1 A n = 2 L Z L 0 n x n 2 2 t/l 2 )sin (4) L n x u 0 (x)sin dx. (5) L ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 41

22 Example: Decay of u 0 (x) =sin( x/l) We will use the following problem to test our finite-di 0 apple x apple L, t 0 (6.a) u(0,t)=u(l, t) =0 (6.b) x u(x, 0) = sin. L (6.c) This model problem has a simple solution that is convenient as a benchmark for numerical schemes for the heat equation. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 42 Example: Analytical solution to decay of u 0 (x) =sin( x/l) Substituting Equation (6.c) into Equation (5) gives A n = 2 L Z L 0 x n x sin sin L L dx. The sine function is orthogonal in the following sense Z L ( x n x L/2 if n =1 sin sin dx = L L 0 otherwise so that 0 A 1 =1, A n =0, n =2, 3,... ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 43

23 Example: Analytical solution to decay of u 0 (x) =sin( x/l) Therefore, the analytical solution to the toy problem is u(x, t) =exp! 2 t x sin. (7) L 2 L The solution is just an exponential decay of the initial condition for this problem and problems with Dirichlet BC and no source term. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 44 Example: Analytical solution to decay of a square pulse 8 >< 0, if 0 apple x<x l u 0 (x) = u c, if x l apple x apple x r >: 0, if x r <xapple L. u c 0 x l x r L An infinite number of A n terms are needed to match the initial condition. A n = 2 Z x r n x u c sin dx L L x l = 2u apple c cos n n =0, 1, 2,... n xl L n xr cos L ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 45

24 Example: Analytical solution to decay of a square pulse t = 1.0e 04 t = 1.0e 03 t = 1.0e 02 t = 5.0e 02 T(x,t) x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 46 Numerical Model Continuous PDE for u(x,t) Finite difference or Finite volume or Finite element or Boundar element or... Discrete difference equation Algebraic solution u i k approximation to u(x i,t k ) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 47

25 Finite Di erence Mesh The finite di erence solution is obtained at a finite set of points. These points are called nodes and the network of these nodes is called a mesh or grid. On a uniform mesh, the x-direction nodes are located at x i =(i 1) x, i =1, 2,...,n x, x = L n x 1 (8) where n x is the total number of spatial nodes, including those on the boundary. i = 1 x = 0... i 1 i i 1 x i 1 x i x i i = n x x = L x x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 48 Finite Di erence Mesh Analogous to the x-direction mesh, there are nodes at discrete times. For uniform time steps t k =(k 1) t, k =1, 2,...,n t, t = t max n t 1 (9) where n t is the number of time steps. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 49

26 Finite Di erence Mesh The x and t direction nodes form a finite version of a semi-infinite strip t k+1 k k Interior node: u(x,t) is initially unknown. Boundary node: u(0,t) and u(0,t) are known or computable from additional data. Initial condition: u(x,0) must be specified. t=0, k=1 i=1 i 1 i i+1 n x x=0 x=l ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 50 Nomenclature For the one-dimensional mesh, we define Symbol Meaning u(x, t) Analytical solution (true solution). u(x i,t k ) Analytical solution evaluated at x = x i, t = t k. u k i Approximate numerical solution at x = x i, t = t k. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 51

27 First order forward finite di erence Use a Taylor series expansion about the point x i u(x i + x) =u(x i xi + x2 2 2 x i + x3 3 3 x i + where x is a small distance, and x 2 and x 3 are shorthand for ( x) 2 and ( x) 3. Solve = u(x i + x) u(x i xi x x 2 u x 2 3! x 3 3 x i + In the limit as x! 0 the coe cients of the higher order derivative terms vanish, and the first term on the right hand side becomes equal to the derivative on the left hand side. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 52 First order forward finite di erence Replace x with x = x i+1 x i,andsubstituteu i u(x xi u i+1 x u i 2 2 u x 2 3! x 3 3 x i + (10) The mean value theorem can be used to replace the higher order derivatives (exactly) 2 2 u + ( 2 3! x 3 3 x i + = where x i apple apple xi u i+1 x u i + x 2 2 u i+1 u xi x (11) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 53

28 First order forward finite di erence In general, is not known. Since x is the only parameter under the user s control that determines the error, the truncation error is simply written = O( x) This expression means that the left hand side is bounded by a product of an unknown constant times x. Although the expression does not give us the exact magnitude of ( x/2) (@ 2 u/@x 2 ), it indicates how quickly that term approaches zero as x is reduced. Recall that the point of Big O notation is to look at the exponent of the truncation error term. We conclude that the error in the first order forward di erence formula is linear in the mesh spacing. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 54 First order forward finite di erence Using big O notation, Equation (10) xi = u i+1 x u i + O( x). (12) Equation (12) is called the forward di erence formula xi because it involves nodes x i and x i+1. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 55

29 First order backward finite di erence Repeat the preceding analysis with a Taylor series expansion for u(x i x) u i 1 = u xi + x2 2 u ( x) 2 x i 3 3 x i + (13) Solve xi to = u i u i u ( x) xi x 2 3! x 3 3 x i + = u i u i 1 + O( x). xi x This is called the backward di erence formula because it involves the values of u at x i and x i 1. ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 56 First order central di erence Combine Taylor Series expansions for u i+1 and u i 1 to = u i+1 u i 1 + O( x 2 ) xi 2 x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 57

30 Summary of first order di erence xi = u i+1 x u i + O( x) Backward: = u i u i 1 + O( xi = u i+1 u i 1 + O( x 2 xi 2 x ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 58 Second Order Central Di erence The Taylor series expansion for u i+1 is u i+1 = u i xi + x2 2 u + ( 2 3! x 3 3 x i + (16) Adding Equation (13) and Equation (16) yields u i+1 + u i 1 =2u i + x 2 2 x i + 2 4! 4 4 x i + Notice that all odd derivatives cancel because of the alternating signs of terms on the right hand side of Equation (13). ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 59

31 Second Order Central Di erence Solving 2 x i gives 2 u = u i 1 2u i + u i x 2 6 x 4 4 x i 2 2 x i = u i 1 2u i + u i+1 x 2 + O( x 2 ). (17) ME 448/548: Introduction to Finite Di erence Approximation of the Heat Equation page 60