# Explicit methods are suitable for large scale

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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 ( + - + + + + - + - + = - + + -

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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
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 ( + - + + + + - + - + = - + + -

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20 Excel for Solving Parabolic PDEs using Finite Difference Equations
21 USING EXCEL Ideal for solving Finite Difference Equations i i+1 i-1 j j-1 j+1 TIME DISTANCE (X)

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22 Explicit Formulation No need for iteration. One press of F9 solves equations
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)

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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]
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