MA557/MA578/CS557. Lecture 14. Prof. Tim Warburton. Spring

Transcription

1 MA557/MA578/CS557 Lectue 4 Sping 3 Pof. Tim Wabuton timwa@math.unm.edu

2 Matlab Notes To ceate a symbolic vaiable (say theta) use the command: theta = sym( theta ); Example manipulation:

3 Matlab cont To evaluate a symbolic function use eval ) Ceate an aay of symbolic functions ) Set theta =. 3) Evaluate the aay of symbolic functions at theta 4) Repeat fo theta=. 3

4 Matlab cont We can also compute Taylo seies expansions fo symbolic functions defined in Matlab: 4

5 Matlab cont An altenative way to compute the coefficients in the Taylo expansion: We obtain the fist 5 coefficients by applying symbolic diffeentiation epeatedly. [Recall: 3 3 δ d f δ d f δ d f f ( δ ) = f ( ) + ( ) + ( ) + 3 ( ) +...! dx! dx 3! dx ] 5

6 Systems of Fist Ode PDEs So fa we have esticted ouselves to teatment of the fist ode advection pde: ρ ρ + u = t x We deived this by a consevation pinciple fo the flow of a fluid (density ho) though a D pipe. 6

7 Lineaized Eule Equations The following pai of equations can be deived fo the pessue and velocity popeties of a gas undegoing small petubations about a zeo mean velocity: p u t x u p t x 7

8 Linea SystemLinea System p p t u x u Just looking at this it is not so clea as to which way the taveling waves go We will apply a multi-dimensional upwinding 8

9 Chaacteistics of the Matix has eigenpais:,,, λ - λ v = = v = = 9

10 Change of Basis We ceate an othonomal tansfomation matix S, with each ow being one of the eigenvectos: S = We ceate the chaacteistic vaiables: q ρ ρ + u = = u ρ u

11 Changing the PDE p p t u x u q q S t + = S x q q t x + S S =

12 Simplification The long way: S S = = = = The shot way: the S matix is the diagonalizing matix

13 Simplification cont q q t x + S S = q q t x i.e. the coupled equations fo ho and u decouple into two independent equations fo: q= p+ u = p u ( ) ( ) q q dt dx = dt dx 3

14 Solution Pocedue ) We stat with u and ho at time= ) We compute q x p x u x ( ) ( ) = ( ) + ( ),,, ( ) ( ) = ( ) ( ) x, p x, u x, 3) We mach in time (to time T): q q t x t x = 4) We back out solutions with: p x T q x T x T u x T q x T x T ( ) (, ) = (, ) + (, ) ( ) (, ) = (, ) (, ) 4

15 DG Scheme We now constuct a DG scheme fo q and a second fo q dx M = L ( x ) L ( x ) dx d q j M Dq j Fq j Gq j dt d j M Dj Hj Jj dt D F G H J nm n n nm n ( ) ( ) ( ) ( ) () () () ( ) nm n m nm n m nm n m dl n L ( x ) ( x ) dx dx = L L = L L L L = L L nm n m + = = + + = 5

16 Numeical Appoach Evaluate chaacteistic vaiables at t= Fo j=:ncells end n= p jm, = mn n = q V n= p jm, = mn n = V ( p( x ) ( )) jn,, + u xjn,, ( p( x ) ( )) jn,, u xjn,, 6

17 Numeical Appoach Time mach using a 3 d ode Runge-Kutta in time and DG in space fo tstep=:ntsteps end q = q, = fo k=3:-: fo j=ncells:-: end end ( ) j j j j j ( ) j j j j j q= q, = dt k dt k q q M Dq + Fq + Gq + M D + H + J 7

18 Finish up Numeical Appoach % evaluate solution at nodes Fo j=:ncells Fo m=:p p p end end jm mn n =,, = = n= p n= p V V jm mn n = ( q ) jn, + jn, ( q ) jn, jn, 8

19 Can We always Get Away With A Chaacteistic Teatment??? Let s ty the following pde: f f t g x g The eigenpais ae now: has eigenpais: v =, λ = i, =, λ = i i v i 9

20 Waning Bells Should Be Ringing v =, λ = i, =, λ = i i v i i S = i define chaacteistic vaiables: q α α i β = S = β α + i β

21 End of This Appoach q q t S S x q i q t i x So the touble is that the solutions will look like: q= q it x ( ) = it+ x ( ) i.e. the chaacteistics move in imaginay time

22 Summay Fo the linea fist ode pdes to be solved as time evolution poblems we equie the matix to have eal eigenvalues (i.e. be hypebolic) We poceeded by decomposing the solution into chaacteistic components which satisfy decoupled fist ode pdes. Fo each chaacteistic component we solved the pde using a DG scheme with upwind fluxes. Next class we will look at using Lax-Fiedichs fluxes with DG.

23 Q) A)What ae the wave speeds fo: Class Poblems B) Is it a hypebolic pde? C) Can we use the upwind DG scheme? q + q t x Q) A)What ae the wave speeds fo: B) Is it a hypebolic pde? C) Can we use the upwind DG scheme? q q t x Q3) How about the second ode wave equation: (hint: intoduce an auxiliay vaiable to educe to a set of fist ode pdes) u u = t x 3

24 Q) a)what ae the wave speeds fo: b) Is it a hypebolic pde? c) Can we use the upwind DG scheme? A) a)wave speeds ae,3,-, b) yes all the eigenvalues ae eal c) yes the system is hypebolic.. Class Poblems q + q t x 4

25 Class Poblems Q) a) What ae the wave speeds fo: b) Is it a hypebolic pde? c) Can we use the upwind DG scheme? A) a) the wave speeds ae: i i q q t x b) no two eigenvalues ae complex c) no 5

26 Class Poblems Q3) How about the second ode wave equation: (hint: intoduce an auxiliay vaiable to educe to a set of fist ode pdes) u u = t x A3) u p = t x p u = = t t x t t x x = t x u p u p u Use the esults fom pevious slides to DG discetize these equations 6