Hyperbolic Partial Differential Equations - Lecture Notes | CSI 701, Study notes of Computer Science

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Hyperbolic PDE’s

1.^

Analysis of numerical schemes

2.^

Finite difference methods

3.^

Finite volume methods

PDE solvers

In general the solution of PDE’s proceeds as follows: 1.

Construct a spatial grid to represent continuousfunctions

Build approximations to the spatial derivatives on thisgrid (spatial discretization)

Obtain a system of ODE’s

Construct a temporal FD scheme to solve the ODEsystem

Advance the solution in time

Example: transient heat equation

Construct the FD temporal scheme to solve the ODEsystem:

Code it and solve it

Remark 1: you will need to estimate the allowable time-step

for your solution scheme

Remark 2: you will also need to build approximations of the

boundary conditions and take them into account in thesolution algorithm

(^

1

1 2

1

n i n i

n i

n i

n i^

u

u

u

t h

u

u^

−

+^

Explicit and implicit schemes

-^

Typically time-dependent PDE’s are first discretized inspace using any of the methods described before

-^

This results in a system of coupled ODE’s that is thendiscretized usually using finite differences

-^

Numerical schemes are then classified as explicit orimplicit depending on how the time derivatives arediscretized

Mixed schemes

-^

Hybrid methods are obtained by linear combination ofexplicit and implicit schemes:

-^

: Forward Euler (explicit)

-^

θ=1/2: Crank-Nicholson (hybrid)

-^

: Backward Euler (fully implicit)

implicit

licit

t^

u

u

u^

exp

,

θ

θ^

(^

)^

(^

1 ) 1

1

(^11) 2

1

1 2

1

2

2

) (^1) (^

• −

−

+^

−

Δ Δ

−

Δ Δ

−

−^

n i

n i

n i

n i n i

n i

n i

n i^

u

u

u tk x

u u

u tk x

u

u

Advancing explicit systems

(^

)

2

2

1 2 3

1 1 2 1 3 1 1

1 2 2 1 2 1

1

1 2

with

form

matrix

scheme

updating

scheme

explicit

tk x

b tk x

a

u u u u b a

a b a

a b a

b a

u u u u

u tk x

u tk x

u tk x

u

u u

u tk x

u

u

n n n n m

n n n n m

n i

n i

n i

n i

n i n i

n i

n i

n i

−

−

M

L

M M M M M M M

L L L

M

Explicit vs Implicit

-^

Explicit schemes require matrix multiplication

-^

Implicit schemes require matrix invertion (solution ofalgebraic system at each time step)

-^

So, why use implicit schemes? …

n

n

n

n

u

u A

u A

u

=

⋅

⋅ = +

1 1 :

scheme

implicit

:

scheme

explicit

Analysis of numerical schemes

Consistency •^

Approximation

PDE for

t,^

Δx

Stability •^

Long term effects of local and round-off errors

Convergence •^

Approximation

exact solution for

t,^

Δx

Accuracy •^

Magnitude of local errors

Efficiency •^

CPU and storage vs accuracy

Consistency

-^

Consistency expresses that the discretized equationsshould tend to the differential equations when

t and

x

tend to 0

-^

Condition on structure of numerical formulation

-^

Discretized equation

differential equation

Consistency: example

(^

)

(^

)^

lim

ion

Approximat

1D

in

Laplacian

, 2 1 1 2 , 0

, 4 , 3 , 2 , 1 , 4 , 3 , 2 , 1

1

1

,^2

⎡^ ⎢ ⎣

−

→Δ

• −

−

L

L L

xxxx

i i

i

xx

x

xxxx

xxx

xx

x

i

xxxx

xxx

xx

x

i

i i

i xx

u x u u u x u

u x u x u x

ux

x u x x u u

u x u x u x

ux

x u x x u u

u u

u u x

In general, not difficult to prove

Accuracy: example

(^

)

(^

)

accurate

order

Second

error of term

Leading

series

Taylor

ion

Approximat

1D

in

Laplacian

, 2

, 2 1 1 2 ,

1

1

,^2

−

−

xxxx

xxxx

i i

i

xx

i i

i xx

u x

u x u u u x u

u u

u u x

L

Stability

-^

The numerical scheme should not allow errors to growunbounded, i.e. amplified without bound between twosteps

-^

Condition on solution of numerical scheme

-^

Numerical solution

exact solution of discretized

equation

Convergence

-^

The numerical solution should approach the exactsolution of the differential equation as

t and

x tend to

zero

-^

Condition on solution of numerical scheme

-^

Numerical solution

exact solution of differential

equation

Lax equivalence theorem

-^

For a well-posed initial value problem and a consistentdiscretization scheme, stability is the necessary andsufficient condition for convergence:

consistency & stability

convergence

Analysis of numerical schemes: 1.

Analyze the consistency condition: this leads to thedetermination of the order of accuracy of the schemeand its truncation error

Analyze the stability properties: this leads to detailed onthe frequency distribution of the error

From these two steps convergence can be established