Introduction to Numerical Methods for Hyperbolic Conservation Laws | MATH 593, Study notes of Mathematics

Download Introduction to Numerical Methods for Hyperbolic Conservation Laws | MATH 593 and more Study notes Mathematics in PDF only on Docsity!

Numerical Methods for Hyperbolic

Conservation Laws

Introduction

Dr. Aamer Haque

http://math.iit.edu/~ahaque

ahaque7@iit.edu

Illinois Institute of Technology

June 2, 2009

Outline



Introduction



Mathematical/Numerical Modeling



Derivation of Conservation Laws



Examples of Conservation Laws



Hyperbolic Conservation Laws

V&V - Verification & Validation

Verification – The process of determining that a model

implementation accurately represents the developer's

conceptual description of the model and the solution to

the model. (i.e. Solving the equations correctly)

Validation – The process of determining the degree to

which the model is an accurate representation of the

real world from the perspective of the intended uses

of the model. (i.e. Solving the correct equations)

Source: AIAA Guide for the V&V of CFD Simulations

Methods of V&V



Verification



Software engineering



Analytical solutions



Sod, Noh, Sedov



Benchmark numerical solutions



Collela-Woodward interacting blast waves



Conservation checks



Total energy



Validation



Experimental data

V&V requires quantitative metrics for comparisons

Conservation Laws for Open Systems

An open system can exchange physical quantities

with its surroundings or with other systems (i.e. Flux)

We also allow the possibility of internal sources/sinks

∫

t

1

t

2

f

in

dt −

∫

t

1

t

2

f

out

dt 

∫

t

1

t

2

∫

V

q dV dt

f

in

f

out

q

∫

V

 r ,t

2

 dV −

∫

V

 r ,t

1

 dV

Control Volume

A control volume V is a connected subset of space

of finite volume bounded by a surface S with finite area

We shall assume that the control volume is fixed in space (Eulerian)

∫

V

r ,t

2

 dV −

∫

V

r ,t

1

 dV =−

∫

t

1

t

2

∫

S

F ⋅ n  dSdt 

∫

t

1

t

2

∫

V

q dVdt

n 

F

V

S

Continuity Equation

Many applications in continuum mechanics have

their flux defined by the following equation:

When the source term is zero, we obtain the

Continuity Equation

F =

v

∂ t

v = 0

e.g. Mass conservation:

Scalar Linear Advection Equations

Linear Advection Equation - Describes the transport of a

quantity u in 1D at constant velocity c

∂ u

∂ t

 c

∂ u

∂ x

Advection-Diffusion Equation - Describes the transport of a

quantity u in 1D with constant velocity c and constant

diffusion coefficient D

∂ u

∂ t

 c

∂ u

∂ x

= D

2

u

∂ x

2

Berger's Equation

Simple 1D model of momentum transport in a fluid

ignoring: viscosity, density/pressure variations

∂ u

∂ t

 u

∂ u

∂ x

∂ u

∂ t

∂ x

u

2

Fokker-Plank Equation

Describes the evolution of the velocity distribution function

The probability density of a particle having a velocity of 

v

f  

v 

∂ f

∂ t



v

⋅[− f  v ]= D ∇



v

2

f.

Applications to plasma physics and astrophysics

Conservation of Mass

Mass cannot be created nor destroyed within a

control volume

We simply have the Continuity Equation

∂ t

v = 0

Some Vector/Tensor Calculus

Index/Einstein summation specifies that:

comma denotes differentiation & repeated indicies are summed

[ ∇ ]

i

,i

[ ∇

v ]

ij

= v

i , j

v = v

i , i

[ ∇⋅ A ]

i

= a

ij , j

v ⋅ n = v

i

n

i

[

v × 

v ]

ij

= v

i

v

j

[ ∇⋅ pI ]

i



p 

ij



, j

=∇ p

Gradient

Divergence

Products

Conservation of Momentum

Change in Momentum = Momentum Flux + External Forces

∂ t

v ∇⋅[ 

v × 

v  pI ]= 0

d

dt

∫

V

v dV =−

∫

S

v ⋅ n 

v dS −

∫

S

p n  dS

Conservation of Energy

Change in Energy = Energy Flux + Work Done

∂ t

 E ∇⋅

[

 E  p  

v ]

d

dt

∫

V

 E dV =−

∫

S

 E

v ⋅ n  dS −

∫

S

p 

v ⋅ n  dS