PARTIAL DIFFERENTIAL EQUATIONS (PDE S) I

Transcription

1 PARIAL DIFFERENIAL EQUAIONS (PDE S) I Numerical methods in chemical engineering Edwin Zondervan

2 OVERVIEW We will find out how we can compute numerical solutions to partial differential equations. We will see the standard types of PDE s (elliptic, paraolic, hyperolic) he approach presented in this lecture to solve a PDE is y reducing the system to a system of ODE s or algeraic equations (rememer the lecture on iterative methods )

3 YPES OF PDE S For a general nd order PDE: a u t u u c t We classify the PDE as follows: ac < Elliptic ac = Paraolic ac > Hyperolic f u t, u, u (7-) 3

4 EXAMPLES Paraolic: (7-) t k C p Space is a two way coordinate, and time is a one way coordinate *meaning that we can march a solution forward in time) Elliptic: (7-3) y and y are oth two way coordinates (we looked at this prolem in the lecture on iterative methods Hyperolic: (7-) u t c u his is the wave equation. We will not discuss this type of PDE

5 SOLVING A PDE NUMERICALLY o solva a PDE, we must first convert it to a system of algeraic equations or ODE s, to accomplish this the domain of the solution must e discretized. here are three methods to discretize the spatial domain: Finite differences (iterative methods and this lecture) Finite volumes Finite elements 5

6 HE MEHOD OF LINES In the method of lines, we discretize that spatial domain of the PDE, to produce a set of ODE s, which govern the temperature at each point in the solution. We will use finite differences to accomplish this. 6

7 OUR PREVIOUS PROBLEM he unsteady heat equation: y Where = t is the thermal diffusivity. (7-5) In two dimensions this equation can e written as: = 3 = = We re going to solve this prolem on the domain defined aove t y (7-6) 7

8 RACK EMPERAURE ON A GRID j =Ny (Ny-)N+ Inde of a node is given y: K = i + N(j-) j =3 So that: j = N+ N+ N+3 N i,j = k=i+n(j-) j = 3 N i = i = i =3 i =N 8

9 GRID NODES We will discretize our heat equation to find an ODE which governs the temperature at node k. d dt i, j i, j y i, j (7-7) 9

10 ESIMAES OF HE DIFFERENIALS Assume a piece-wise linear profile in the temperature: i+ i / i / i- i i i, j, j i, j i, j i i, j, j i, j

11 ESIMAES OF HE DIFFERENIALS If we make a similar approimation in the y direction, we can write the rate of change of temperature at node k as: d dt d dt i, j k i, j k k i, j k i, j k N i, j y k i, j y k N i, j (7-8) (7-9)

12 HE BOUNDARY CONDIIONS he oundary nodes have a fied temperature, so they do not fullfil our discretized equations. here are several ways to include the oundaries into our solution approach: Eliminate the temperatures at oundary nodes from the set of equations Write the oundaries at d oundary /dt = Include the oundary node equations as a set of algeraic equations, which must e solved parallel to our set of ODE s

13 ELIMINAE BOUNDARY NODES 3 j =Ny (N(Ny-)+ (N(Ny-)+ N j = N+ N j = N+ N i = i = i =N 3

14 NODES ON A BORDER For a node on a order counts (when = y): d dt k ( k k k N k k N ) (7-) Node on order : k-n = Node on order : k+n = Node on order 3: k- = 3 Node on order : k+ =

15 5 Numerical Methods HE COMPLEE MODEL dt d A dt d (7-) (7-)

16 HE SOLUION t=.968e-7 tau t=.7 tau t= 3 tau = L / time constant (gives an idea on how long we should run the simulation, L is the characteristic length scale for condution: L = (N+) /)

17 SABILIY We saw earlier that staility of a system can e evaluated y checking the Jacoian; the eigenvalue of the Jacoian with the largest magnitude determines which time step you can take! For our system, the Jacoian is simply given y: J = A 7

18 GERSHGORIN S HEOREM For a square matri and row k, an eigenvalue is located on the comple plane within a radius equal to the sum of the moduli of the offdiagonal elements of that row. m m m... m m m k, k k, k, k, N k, N k, N (-6) Im( ) Radius = sum of moduli of off-diagonal elements he eigenvalue k must e in this circle. m k,k Re( ) 8

19 FOR OUR CASE All eigenvalues should e within a radius of / on an Argand plot: Im( ) Using PDE s with eplicit schemes means that we have a maimum step size we can take in order to otain a stale solution r= / Re( ) 9

20 SUMMARY PDE s can e converted into a set of ODE s PDE s produce sparce Jacoians he eigenvalues of the Jacoian depend on the grid spacing influencing staility.