Numerical Methods for Partial Differential Equations, Slides of Mathematical Methods for Numerical Analysis and Optimization

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Numerical Methods for Partial

Differential Equations

AB

• The general AB scheme reads:
• To check linear stability we assume that f is linear in u (we can

also achieve a similar goal by assuming f is Lipschitz)

• Assuming:
• We know that all the solutions of this recurrence relationship are

bounded provided the corresponding stability polynomial:

has roots which satisfy the root condition (all roots bounded by 1,

and if they have magnitude 1 must be simple roots).

1 (^1 ) 1

m s

n n m n m m

u u dt α f u

=

• • − =

= + (^) ∑

( ) (^1 ) 1

m s

n n n n m n m m

f u μu u u dt μ α u

=

• • − =

= ⇒ = + (^) ∑

1

1

m s s s s m m m

z z dt μ α z

= − −

=

= + (^) ∑

AB cont

• Given the root equation:
• We determine the map in a trivial manner:
• Example AB3:

1

1

m s s s s m m m

z z dt μ α z

= − −

=

= + (^) ∑

1 1

1

1

m s^ s^ s s s s m m (^) m s m s^ m m m

z z

z z dt z dt

z

μ α μ

α

= − − − = = −

=

= + (^) ∑ ⇒ =

∑

3 2

z z

z z

ν

AB3 Stability

3 2

z z

z z

ν

i

z re

θ

ν

cont

• We now use Taylor expansions to express the

truncation error in terms of u and its derivatives

at tn:

• We found that Tn scaled as:
• Finally we apply Dahlquist’s equivalence

theorem, which requires consistency and stability

 convergence and corollary which gives us

global error estimates:

( (^1) ) ( ) ( ( 1 )) 1

m s

n n m n m n m

u t u t dt α f u t T

=

• • − =

= + (^) ∑ +

( ) [^ ]

1 1 * *

1 for some constant^ and^ ,^1

s s n (^) s n n n n

d u

Cdt t C t

dt

T t t

( ) (^) n, , ,...

s

u tn − un ≤Ct s u dt

RK Analysis

• A similar analysis is required for the Runge-Kutta

schemes.

• The stability analysis was straightforward as we

found that the recurrence for a linear f was:

• For stability we required:
• We examined this by setting

1

m s^ m

n n m

u u

m

ν

=

=

∑

0

m s^ m

m m

ν

=

∑ ≤

m s m i

m

e

m

ν θ

=

=

∑ =

RK Accuracy

• The error estimate process for RK schemes is a

little more complicated, but still boils down to

dt expansions of the one-step operator.

• Again, once we have consistency and stability

we have convergence.

Implicit Schemes

• Later on in the semester we will discuss

implicit schemes.

• Then we will also discuss the Butcher block

formulation of RK methods.

Sample Points in Time Space

t n

u u c t x

∂ ∂

∂ ∂

• We chose to sample the ODE at discrete points on the time axis.
• We can also make the same choice for the spatial representation

at each discrete time level.

• We follow this choice with the dilemma of how to reconstruct an

approximation to the derivative of discretely sampled data.

xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

Choice of Stencil

t n

xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

t n

xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

t n xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

t n xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

t n xm (^) − 2 xm (^) − 1 x m xm (^) + 1 xm (^) + 2

x xm (^) − 3 xm (^) + 3

And many more combinations Docsity.com

Left Differencing

• Using the data points on this stencil we wish to

compute an approximation to the gradient of u

at

• Dividing and rearranging we find:

t n

xm − 2 xm − 1 x m xm + 1 xm + 2

x

xm − 3 xm + 3

x m

2 2 3 3

(^1 2 )

m m m m m

du dx d u dx d u

u x u x dx x x x

dx dx dx

− =^ −^ +^ −

2 2 3 (^1) * 2 3

m m m m m

du u x^ u x dx d u dx d u

x x x

dx dx dx dx

Left Differencing

t n

xm − 2 xm − 1 x m xm + 1 xm + 2

x

xm − 3 xm + 3

2 2 3 (^1) * 2 3

m m m m m

du u x^ u x dx d u dx d u

x x x

dx dx dx dx

This gives us an obvious one-sided linear interpolation formula

of the derivative.

We will later on decide whether it is suitable for time-stepping with

AB or RK.

Right Difference Operator

t n

xm − 2 xm − 1 x m xm + 1 xm + 2

x

xm − 3 xm + 3

m 1 m m

u u

u

dx

2 2 3

2 3

m m m m

du dx d u dx d u

u x x x

dx dx dx

i.e. delta+ is a first order approximation of the derivative operator

Central Differencing Operator

• Subtracting the forward and backward

expansions:

• We obtain:

2 2 3 3 4 4 5 5

(^1 2 3 4 )

2 2 3 3 4 4 5 5 ** (^1 2 3 4 )

m m m m m m m

m m m m m m m

du dx d u dx d u dx d u dx d u

u x u x dx x x x x x

dx dx dx dx dx

du dx d u dx d u dx d u dx d u

u x u x dx x x x x x

dx dx dx dx dx

−

3 3 5 (^1 1 )

m m m m

du dx d u

u x u x dx x x O dx

dx dx

+ −^ − =^ +^ +

t n

xm − 2 xm − 1 x m xm + 1 xm + 2

x

xm − 3 xm + 3