Review Parabolic PDEs Summary PARABOLIC PDES. Dr. Johnson. School of Mathematics. Semester university-log

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1 PARABOLIC PDES School of Mathematics Semester

2 OUTLINE 1 REVIEW 2 PARABOLIC PDES 3 SUMMARY

3 OUTLINE 1 REVIEW 2 PARABOLIC PDES 3 SUMMARY

4 OUTLINE 1 REVIEW 2 PARABOLIC PDES 3 SUMMARY

5 ELLIPTIC PDES Elliptic equations can usually be written in the form w i+1,j 2w i,j +w i 1,j x 2 + w i,j+1 2w i,j +w i,j 1 y 2 + = 0, The solutioncan then be expressedas the solutionto the matrix equation Ax = b The general iteration scheme can be written as x k+1 = Px k +Q The rate of convergence depends on the spectral radius of the iteration matrix. university-log

6 ELLIPTIC PDES Elliptic equations can usually be written in the form w i+1,j 2w i,j +w i 1,j x 2 + w i,j+1 2w i,j +w i,j 1 y 2 + = 0, The solutioncan then be expressedas the solutionto the matrix equation Ax = b The general iteration scheme can be written as x k+1 = Px k +Q The rate of convergence depends on the spectral radius of the iteration matrix. university-log

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8 EXAMPLES One of the simplest parabolic pde is the diffusion equation which in one space dimensions is u t = κ 2 u x 2. Fortwo or more space dimensionswe have u t = κ 2 u In the above κ issome givenconstant.

9 EXAMPLES Another familiar set of parabolic pdes is the boundary layer equations u x +y y =0, u t +uu x +vu y = p x +u yy, 0 = p y.

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11 INITIAL CONDITIONS For parabolic PDEs we expect, in addition to the boundary conditions,aninitial conditionatsay,t = 0. t R S x university-log

12 HEAT EQUATION Let us consider the heat equation inthe regiona x b. u t = κ 2 u x 2. Take auniformmeshinxwithx j = a +j x, for j = 0,1,...,n and x = (b a)/n. Forthe differencingintime we assume aconstant stepsize t so that t = t k = k t.

13 HEAT EQUATION Let us consider the heat equation inthe regiona x b. u t = κ 2 u x 2. Take auniformmeshinxwithx j = a +j x, for j = 0,1,...,n and x = (b a)/n. Forthe differencingintime we assume aconstant stepsize t so that t = t k = k t.

14 HEAT EQUATION Let us consider the heat equation inthe regiona x b. u t = κ 2 u x 2. Take auniformmeshinxwithx j = a +j x, for j = 0,1,...,n and x = (b a)/n. Forthe differencingintime we assume aconstant stepsize t so that t = t k = k t.

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16 FIRST ORDER APPROXIMATION We may approximate our equation by w k+1 j t w k j [ w k j+1 2w k ] j = κ +wk j 1 x 2. Here w k j denotesanapproximation to the exactsolution u(x,t) ofthe pde atx = x j,t = t k. The above scheme isfirst orderin timeo( t) and second orderinspace O( x) 2. This scheme isexplicitbecause the unknowns atlevelk +1 can be computed directly. university-log

17 FIRST ORDER APPROXIMATION We may approximate our equation by w k+1 j t w k j [ w k j+1 2w k ] j = κ +wk j 1 x 2. Here w k j denotesanapproximation to the exactsolution u(x,t) ofthe pde atx = x j,t = t k. The above scheme isfirst orderin timeo( t) and second orderinspace O( x) 2. This scheme isexplicitbecause the unknowns atlevelk +1 can be computed directly. university-log

18 FIRST ORDER APPROXIMATION We may approximate our equation by w k+1 j t w k j [ w k j+1 2w k ] j = κ +wk j 1 x 2. Here w k j denotesanapproximation to the exactsolution u(x,t) ofthe pde atx = x j,t = t k. The above scheme isfirst orderin timeo( t) and second orderinspace O( x) 2. This scheme isexplicitbecause the unknowns atlevelk +1 can be computed directly. university-log

19 BOUNDARY CONDITIONS Let us assume that we are given a suitable initial condition, and boundary conditions of the form u(a,t) = f(t) u(b,t) = g(t). Notice thatthere is atime lagbeforethe effectofthe boundary data is felt on the solution.

20 STABILITY CONDITION As wewillsee laterthis scheme is conditionallystable for where β 1 2 β = κ t x 2. Note that β is sometimes called the Peclet or diffusion number.

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22 IMPLICIT SCHEME Abetterapproximation is onewhich makes useof the most up-to-date information. Taking our approximations at the k +1timelevelwehave w k+1 j t w k j = κ [ w k+1 j+1 2wk+1 j +w k+1 ] j 1 x 2.

23 IMPLICIT SCHEME Abetterapproximation is onewhich makes useof the most up-to-date information. Taking our approximations at the k +1timelevelwehave w k+1 j t w k j = κ [ w k+1 j+1 2wk+1 j +w k+1 ] j 1 x 2. The unknowns at level k +1 are coupled together and we have a set of implicit equations to solve.

24 SYSTEM OF EQUATIONS Rearrange to get βw k+1 j+1 + (1 +2β)wk+1 j βw k+1 j 1 = wk j, for1 j n 1 Approximation of the boundary conditions gives w k+1 0 = f(t k+1 ), w k+1 n = g(t k+1 ) We have a tridiagonal system of equations.

25 SYSTEM OF EQUATIONS Rearrange to get βw k+1 j+1 + (1 +2β)wk+1 j βw k+1 j 1 = wk j, for1 j n 1 Approximation of the boundary conditions gives w k+1 0 = f(t k+1 ), w k+1 n = g(t k+1 ) We have a tridiagonal system of equations.

26 PROPERTIES OF THE SCHEME We can usedirectmethods to solveatridiagonalsystemof equations. The scheme is only firstorder,the same as the explicit scheme. However it is unconditionally stable - there are no restriction on the magnitude of β.

27 PROPERTIES OF THE SCHEME We can usedirectmethods to solveatridiagonalsystemof equations. The scheme is only firstorder,the same as the explicit scheme. However it is unconditionally stable - there are no restriction on the magnitude of β.

28 FIRST ORDER METHODS FOR PARABOLIC PDES The explicit method is the simplest method, taking the differenceapproximations at t k. The scheme isfirst order in t, The stability condition requires β 1/2. The implicit method takes the difference approximations att k+1 The scheme isfirst order in t, The scheme is unconditionally stable. Likethe modifiedeulermethodfor ODEs,we cantake our differenceequationsat t k+1/2 to increase the orderofthe scheme. Nexttime - secondorderschemes... university-log

29 FIRST ORDER METHODS FOR PARABOLIC PDES The explicit method is the simplest method, taking the differenceapproximations at t k. The scheme isfirst order in t, The stability condition requires β 1/2. The implicit method takes the difference approximations att k+1 The scheme isfirst order in t, The scheme is unconditionally stable. Likethe modifiedeulermethodfor ODEs,we cantake our differenceequationsat t k+1/2 to increase the orderofthe scheme. Nexttime - secondorderschemes... university-log

30 FIRST ORDER METHODS FOR PARABOLIC PDES The explicit method is the simplest method, taking the differenceapproximations at t k. The scheme isfirst order in t, The stability condition requires β 1/2. The implicit method takes the difference approximations att k+1 The scheme isfirst order in t, The scheme is unconditionally stable. Likethe modifiedeulermethodfor ODEs,we cantake our differenceequationsat t k+1/2 to increase the orderofthe scheme. Nexttime - secondorderschemes... university-log