Lecture3 Pss
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Transcript of Lecture3 Pss
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8/11/2019 Lecture3 Pss
1/12
Ordinary Differential Equations (Chapter 8)
=
=
,
= Taylor Series Expansion
+ = + +
! +
! +
Given we can calculate higher order derivatives and substitute in the aboveequation.
Local Truncation Error = +. Global Truncation Error = Eulers method
+ = + ,+ ()
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Runge-Kutta Methods
+ = + +
! +
! +
From the differential equation we have
() =Applying chain rule
= + = ++ + +( +)
Substituting in the top equation we have
+ = + + ++ (
)
We can rewrite this as
+ = + ++ + + (
)
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We can expand
+ ,+ =+ + + ()Giving
+ = + ++ , + +
[]which can be rewritten as
+ = +
+
= , = ( + , + )This is second order RK method known as Heuns method. We can write a general
second order method as
+ = + + + , + + () + = + + + + + []
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Comparing Eqns a and b we get
+ =
= =
Have one less equation than unknowns. If we set = = = = We get Heuns method. If we set
=
,
=
=
=
we get a
modified Eulers method
+ = + = ,; = ( + , +
)
Exercise :
Derive a 3rdorder method.
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Fourth-Order RK method
+ = + + + + + ()Where
= , = +
, +
= + , + = + ,+
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General form of linear multistep methods (LMM)
+
+
. .
=
+
+
. +
If = , method is explicit.Let
= + +
= + + The LMM is convergent if
(1) All roots of p lie in the unit circle and each root of modulus 1 is simple.
(2) p(1) = 0and
=
(
)
Explicit methods are conditionally stable. Can use LMMs for predictor-corrector.
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Higher order differential equations
Let
() =(,, , ,)With initial conditions
= ; = . ; =
Define new variables
= ; = = ()Now we can form the linear system of first-order-ODES
=
= =(,,, . ,)
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1-D unsteady heat/diffusion equation
,
=
,
, = ; , = , =()Let us discretise the above
= ; = + ; = , = ; = , . .
()
Question: How does one write a finite difference formulation?
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1.FTCS (Forward in time and centered in space)
, =
+ +
, =
+
+ =
+ +
+ =
+ +
+
For this method to be stable we need
> 0 <
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This can be written as
(+) = Can use TDMA or Iterative Methods to solve the above. The method is
unconditionally stable.
