Explicit methods are suitable for large scale

Info icon This preview shows pages 17–24. Sign up to view the full content.

View Full Document Right Arrow Icon
Explicit methods are suitable for large scale problems where the phenomena occurs quickly. For example a car crash simulation. In this case, at time t, we do not need to solve a matrix system as: 2 , 1 , , 1 1 , ) 2 ( x t r u u u r u u j i j i j i ij j i = + - + = - + + α Advantage of implicit methods is that we can use larger time steps, but we also need to store and solve a matrix system.as: j i j i j i j i j i j i ru u r ru ru u r ru , 1 , , 1 1 , 1 1 , 1 , 1 ) 2 2 ( ) 2 2 ( + - + + + + - + - + = - + + -
Image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
18 Example Use Crank-Nicholson scheme to find system of equations for time = 0.01.Show that they give the results given in the table below. ICE i=0 1 2 3 4 5 6 0 = x u i=0 1 2 3 4 5 6 x=0 0.1 0.2 0.3 0.4 0.5 0.6 t=0 0 0.2 0.4 0.6 0.8 1 0.8 t=0.01 0 0.1989 0.3956 0.5834 0.7381 0.7681 0.7381
Image of page 18
19 Solvers for Parabolic PDEs ) 2 ( , 1 , , 1 1 , j i j i j i ij j i u u u r u u - + + + - + = Equations resulting from an explicit scheme are easy to solve as the variable at the latest time step is dependent on variables ALL at the previous time step. For example Equations resulting from an implicit scheme require Matrix solvers (Direct or Iterative) as the variable at the latest time also depends on unknown values at that time step. For example For the Implicit method we can use the Gauss Seidel iterative method to solve the matrix system. j i j i j i j i j i j i ru u r ru ru u r ru , 1 , , 1 1 , 1 1 , 1 , 1 ) 2 2 ( ) 2 2 ( + - + + + + - + - + = - + + -
Image of page 19

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
20 Excel for Solving Parabolic PDEs using Finite Difference Equations
Image of page 20
21 USING EXCEL Ideal for solving Finite Difference Equations i i+1 i-1 j j-1 j+1 TIME DISTANCE (X)
Image of page 21

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
22 Explicit Formulation No need for iteration. One press of F9 solves equations
Image of page 22
23 Explicit Scheme Courant Number (r) Parameters: rr (= dt/dx^2) dx dt Time (=A25+dt) Explicit Equation D30=D29+rr*(C29-2*D29+E29)
Image of page 23

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
24 Implicit Scheme Need to use Iterative Methods IF Command IF(CONDITION, THEN DO THIS, ELSE DO THIS) e.g if(flag=0,0,…) FLAG Parameter (1 or 0) Implicit Equation =if(flag=0,0,(rr*B27+(2-2*rr)*C27+rr*D27+rr*B28+rr*D28)/(2+2*rr)) [B28 & D28 are at current time-step]
Image of page 24
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern