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Finite difference methods
An introduction
Jean Virieux
Professeur UJF
2012-2013
with the help of Virginie Durand
A global vision Differential Calculus (Newton, 1687 & Leibniz 1684)
Find solutions of a differential equation (DE) of a dynamic system.
Chaos Systems (Poincaré, 1881) Find properties of solutions of the DE of a dynamic
system. Chaos & Stability (Smale, 1960)
Find properties of solutions of a physical system without knowing its DE
After the presentation of Etienne Ghys, 13 october 2009
This course is about the differential calculus using the finite difference approach familiar to Newton & Leibniz
Bibliography on Finite Difference Methods :A. Taflove and S. C. Hagness: Computational Electrodynamics: The Finite-
Difference Time-Domain Method, Third Edition, Artech House Publishers, 2005
O.C. Zienkiewicz and K. Morgan: Finite elements and approximmation, Wiley, New York, 1982
W.H. Press et al, Numerical recipes in FORTRAN/C …Cambridge University Press, USA, 20XX …
http://en.wikipedia.org/wiki/Finite-difference_time-domain_method
http://en.wikipedia.org/wiki/Upwind_scheme
http://en.wikipedia.org/wiki/Lax-Wendroff_method
Spice group in Europe : http://www.spice-rtn.org
FDTD introduction :
ftp://ftp.seismology.sk/pub/papers/FDM-Intro-SPICE.pdf
By P. Moczo, J. Kristek and L. Halada
What is the main idea?• Coverage of the computational domain by a space-time grid
• Approximate derivatives and initial values at all grid points• Define boundary conditions at end points• Construct the linear algebraic system to be solved by the
computer
The heat equation2
2
( , ) ( , ) ( , )u x t u x tk x tt x
k3,3 10-72 10-7
1,15 10-61,44 10-77,3 10-76,7 10-71,1 10-6k en m2/s
Sol humide (8%)Sol secGlaceEau
CalcaireBasalteGranite
k is the thermal diffusion coefficient
Replace partial derivatives by finite difference approximations leading to an algebraic system
u(x,t) ~ Uin where the index
i is for the discrete spatial position and n for the discrete time level
Other PDE in physics
2
2
( , ) ( , )( )u x t u x txt x
2 22
2 2
( , ) ( , )( )u x t u x tc xt x
2
2
( )( ) u xf xx
The scalar wave equation is a partial differential equation which belongs to second-order hyperbolic system.
2
2
( , )0 u x tx
2 22
2 2
( , ) ( , )( )u x t u x txt x
2
2
( , ) ( , )( )u x t u x txt x
Wave Equation
Fluid Equation
Diffusion Equation
Laplace Equation
Fractional derivative Equation
Time is involved in all physical processes except for the Laplace equation related to Newton law and mass distribution.
Poisson equation could be considered as well when mass is distributed inside the investigated volume
Poisson Equation
( , ) ( , )( ),u x t u x ta xt
tx
Advection Equation
Finite Difference Stencil
i-1 i i+1
(Leveque 1992)
centeredhUUUD
backwardhUUUD
forwardh
UUUD
iii
iii
iii
211
0
1
1
Truncations errors : 0h
Second derivative
iii UDUDDUDD 200
)2(1112
2iiii UUU
hUD
Higher-order terms : same procedure but you need more and more points
h
Finite difference approximation1
1
1 1
21 ( )2
i i
i i
i io
o
U UDh
U UDh
U UDh
D D D
forward
backward
centeredForward/backward first-order approximation
Centered second-order approximation
Higher-order approximations could be considered as well : more values !!!
Second-order accurate central-difference approximation
Leapfrog second-order accurate central-difference approximation
Leapfrog 4th-order accurate central-difference approximation
Stencil length
Duality between a function and its derivative
xux
xux
xux
xuxuuxxu
nininininini 4
44
,3
33
,2
22
,,,1 2462
)(
xux
xux
xux
xuxuuxxu
nininininini 4
44
,3
33
,2
22
,,,1 2462
)(
xux
xuxuuu
nininini 4
44
,2
22
,,1,1 122
Discretisation and Taylor expansion
)(2 2
2,,1,1
,2
2
xx
uuuxu ninini
ni
Assuming an uniform discretisation x,t on the domain, we consider interpolation upto power 4
by summing, we cancel out odd terms
neglecting power 4 terms of the discretisation steps. We are left with quadratic interpolations, although cubic terms cancel out for precision.
Second derivatives2 1 1
2
2 1 1
2
1
i i i
i i i i
U U UDh
U U U UD D D D Dh h h
Various methods for evaluation of second derivatives
Taylor interpolation formulation related to Lagrande polynomials and, therefore, values are not restricted to nodes of a grid … we assume a smooth contribution of the numerical solution.
The solution of the heat equation has a numerical approximate solution
1 1
1 12
1 11 12
1 2
2
n nn n ni ii i i
n
n n n n ni i i i i
U U U U Ut x
tU U U U Ux
Stability: Errors are bounded during the computation of the solution.
Consistence: Truncation errors go to zero when discretisation gets smaller.
Is the numerical solution correct ?S
olution Accuracy Convergence: Numerical solutions go to the exact
solution as discretisation gets smaller.
Heterogeneous mediaWhen considering numerical methods, we often address the problem of complex media for which there is no analytical solution
Spatial derivatives of medium properties should be estimated
Decreasing the order of derivatives by adding auxiliairy variables which have often physical meanings ….
, ,,
, ,
,
Parsimonious approach1
1
1 1
11
2
2
n ni i
n ni
n nni i
i i i
ni
ii
U UR C FV Vt
UV
xUx
K
1 11 2 1 2
1 1 1 1
12
2
1
2
( )4
2 22
n n n n nni i i i i i i i i
i i
n n n ni i i i
i
n n i ini i
i i i
U
K KU UR C
U K U K U U K KR C F
U U U Ux x
xF
t
t x
Standard grid
Not exactly waht we want because of indices i+2 and i-2: we need to move to a mixed grid approach (duality between a function and its derivative)
Parsimonious approach1
1/ 2 1/ 21
1/ 2 1/ 21
n nn nni i
i i i
ni
i i
n ni
ii
U UR C Ft
V
V Vx
U Ux
K
1 11/ 2 1 1/ 2 1 1/ 2 1
1 1 1/ 2 1
/ 22
/1
21
( )
n n
n n i ini i
n n n n nni i i i
n
i i i
ni i i
i i i i ii i i
i
U
U U U Ux x
x
K KU U
U K U K U U K KR C
F
Ft x
R Ct
Staggered grid
or mixed grid approach
Same indices than the homogeneous case: arithmetic average coming from the physics behind …
Identify how to define jumps at an interface
Time marching procedure
1 11 12 2n n n n n n
i i i i i itU U U U U
x
i-1 i i+1
n-1
n
n+1
A very simple construction: estimation of a future value from present values and one past value.
Initial values:
Boundary values:
1 0iU
1 0 & 0n nLU U Dirichlet conditions
IS THAT ALL ?
Explicit centered scheme
Frequency approach
Finite Difference formulation leads to a linear system
1/ 2 1 1/ 2 1 1/ 2 1/ 22
( )i i i i i i ii i i i
K U K U K K Ui R CU Fx
WRITE ON THE BLACKBOARD THE LINEAR SYSTEM
FIND A SOLUTION ?
In Time : marching approach
In Frequency : linear algebra
ODE versus PDE formulations
GOAL : find ways to transform differential operators into algebraic operators in order to use linear algebra at the end
( ) ( ( , ), , )
( ) ( , ) ( , )
d y t A y x t x tdtd y t A x t y x tdt
( , ) ( ( , ), , )
( , ) ( , ) ( , )
y x t D y x t x tt
y x t D x t y x tt
O.D.E
Ordinary differential Equations
P.D.E
Partial Differential Equations
Linear
Non-linear
Symmetry between space and time ?
« simple » solution « complex » solution
EXISTENCE & UNIQUENESS
The numerical problem is well posed usingHadamard criteria if the solution exists, isunique and depends continuously to initial/boundary conditions, to boundaries and coefficients of the differential equations
Von Neumann stability analysis for FD methods: numerical integration cannot go beyond physicaldistances of interaction.
For example, Courant-Friedrichs-Lewy CFL condition for wave propagation
First-order hyperbolic equationuEx
uvt
xv
t
xc
tv
2
xuE
xtu
2
2
Let us define other variables for reducing the derivative order in both time and space
The 2nd order PDE became a 1st order PDE
This is true for any order differential equations: by introducing additionnal variables, one can reduce the level of differentiation. Among these different systems, one has a physical meaning
which becomes
xvE
t
xtv
1
Ec 2
with
stress
velocity
Other choices are possible as displacement-stress instead of velocity-stress.
Initial and boundary conditions
Boundary conditions u(0,t)
Initial conditions u(x,0)
Boundary conditions u(L,t)
1D string medium
Difficult to see how to discretize the velocity c(x) !
f(x,t)=s(t)r(x) Excitation condition
Dirichlet conditions on u
Neumann conditions on
, ,
Much better for handling heterogeneities, ,
, ,
v=0
Source excitationImpulsive source: Ricker wavelet for example
1 2
Oscillatory source: Sinus wavelet for example
sin 22
Source radiation Directional source (hammer) Explosive source (dynamite)
Meaning for the string?
Application of of opposite sign forces on two nodes or a fictiousforce between two nodes
Simulation in a two-layer medium
c1=2000 m/s c2=4000 m/s - S=1 in the high-velocity layer
Radiation boundary condition
Code df1d.3.f
Seismic propagation in the Angel Bay nearby Nice (France)
Earthquake of magnitude 4.9 at a depth of 8 km
CONCLUSION Efficient numerical methods for
propagating seismic waves Time integration versus frequency
integration Competition between FE & FV for
modelling FD an efficient tool for imaging
THANKS YOU !