Diapositiva 1

1
Numerical methods to solve parabolic PDEs
2
Mathematical models 5 Classification

• Classification based on the type of the solution
  (except, usually, the empirical models)
• ANALYTICAL SOLUTIONS ? MATHEMATICAL ANALYSIS?
  ALGEBRA
  
• NUMERICAL SOLUTIONS? NUMERICAL COMPUTATION

• Ex. numerical methods for PDE
• FINITE DIFFERENCE METHOD (FDM)
• FINITE ELEMENT METHOD (FEM)
• METHODS OF LOCATION

3
Partial differential equation (PDE)

• Second order Partial differential equation
• where uf(x,y,t), a function that satisfies this
  equation and that is at least two times
  differentiable, is said to be a solution of the
  equation.
• To obtain a unique solution, it is necessary
  supply some additional information, namely
  initial (IC) and boundary (BC) conditions.
• BC must to be of the order N-1 if N is the order
  of the equation.

4
Partial differential equation (PDE)

• Classical classification for 2nd order PDEs

if ? b2 - 4ac lt 0 Elliptic 0
Parabolic gt 0 Hyperbolic
5
Partial differential equation (PDE)

• Classical classification for 2nd order PDEs

Laplaces equation for heat conduction
a1, b0, c1 ? b2 - 4ac -4 Heat
conduction equation is elliptic.
6
Partial differential equation (PDE)

• Classical classification for 2nd order PDEs

Molar Diffusion

• aD, b0, c0
• b2 - 4ac 02 - 4?D ?0 0
• Diffusion equation is parabolic.

7
Partial differential equation (PDE)

• Classical classification for 2nd order PDEs

Convection in conservation laws
First order ? ?/?t or ?/?h
a1, bv, c0 av, b1, c0 ? b2 - 4ac v2
? b2 - 4ac 1 Convection equation is
hyperbolic
8
Second order Partial differential equation
Classification
Type Equation Description
Parabolic Unsteady-state problem in which transport by conduction or diffusion is important
Elliptic Steady-state problem in which transport by conduction or diffusion is important
Hyperbolic Unsteady-state problem in which transport by convention phenomenon is important
Note An equation can belong to a class in a
certain field and another in a different field
9
Parabolic PDEs
Parabolic (or diffusion) PDE with one spatial
independent variable
2 order PDE 2 independent variables Linear with
constant coefficients ? is a diffusion coefficient
t ? 0, 8 x ? xA, xB I.C. u(t0,x)u(x)
all x B.C.1 u(t, xxA)uA(t) all t (simple
case) B.C.2 u(t, xxB)uB(t) all t (simple
case) Analytical solution uu(t,x) where u is a
continuous function of t and x
10
Analytical vs numerical solution
Analytical solution The region described by these
two independent, continuous variables (t and x)
is a part of a plane, for the problem under
consideration the length variable, x, varies
between xA and xB, and the time variable, t,
increase without limit from 0. Numerical
solution To obtain a numerical solution , one
replaces these continuous variables with discrete
variables. When these two continuous independent
variables are replaced by discrete variables
(also called x and t), the new variables are
defined at specific points located in the same
region as shown in Figure. The relations between
these discrete variables are finite differential
equations, and it is these finite differential
equations which are solved numerically on a
digital computer.
uu(t, x)
The index i indicates the position along the
x-axis, and the index n is used for the t-axis
uu(tn, xi)
11
General procedure for solving Parabolic PDE
equations
The dependent variable, u, is now a function of
two discrete independent variables, x and t. It
is therefore necessary to use two subscripts to
specify the value of u at a given point
thus u(xi, tn)uin
12
General procedure for solving Parabolic PDE
equations
u(xi, tn)uin The value of the dependent
variable is unknown at a row of points at each
time level, and there are actually an unlimited
number of time levels. It is not feasible to
solve for all the unknown values of u
simultaneously even when a limited number of time
level are considered. Consequently the
technique employed is to solve for the unknown
values of u at one time level, using the know
values of u at the previous time level.
13
General procedure for solving Parabolic PDE
equations
The values of u at the time level when n0, are
given by the initial conditions of equation
(IC). These value are used to determine the
unknown values of u at the next time level for
which n1. The same computational values is then
used to find the values of u for n2 from the now
known values of u at n1, and so on.
I.C.
I.C.
14
General procedure for solving Parabolic PDE
equations
This procedure is continued for as many time
increments as desired. Therefore, the finite
difference equations are formulated so that they
contain values of u at two consecutive levels.
The index n is used to designate the last time
level at which the value of u are known, and the
index n1 is used to designate the next time
level at which variables of the dependent
variable are unknown.
unknown value unless it is given by a boundary
condition. known value
15
Finite Difference Methods
The finite difference method (FDM) was first
developed by A. Thom in the 1920s under the
title The method of square to solve nonlinear
hydrodynamic equations. A. Thom and C. J.
Apelt, Field Computations in Engineering and
Physics. London D. Van Nostrand, 1961. The
finite difference techniques are based upon the
approximations that permit replacing differential
equations by finite difference equations. These
finite difference approximations are algebraic in
form, and the solutions are related to grid
points.
16
Finite Difference Methods

• Thus, a finite difference solution
• basically involves three steps
• Dividing the solution into grids of nodes
• Approximating the given differential equation by
  finite difference equivalence that relates the
  solutions to grid points (In finite difference
  methods, each derivative of the PDE is replaced
  by an equivalent finite difference approximation)
• Solving the finite difference equations subject
  to the prescribed boundary conditions and/or
  initial conditions

17
Finite Difference Methods
If xA0 and xBL, the region between 0 and L
along the x-axis is divided into N equal
increment of size Dx, whit grid points being
placed on each boundary. The time-axis is
divided into increments of size Dt. tnnDt
xixAiDxiDx Some useful relations
between values of the independent variable at
adjacent points xi1xiiDx xi-1xi-iDx For
many numerical solutions, it will be desirable to
increase the size of the time step, Dt, as the
solutions progresses.
18
Finite Difference Methods
Both sides must to be evaluated at the same
conditions (xi and tn)
We will indicate
Evaluate at time tn
Evaluate at length xi
All i and n
The FDM write both sides and check the equality
for each grid points
19
Finite Difference Methods

• The basis of a finite difference method is the
  Taylor series expansion of a function.
• Consider a continuous function f(x). Its value
  at neighboring points can be expressed in terms
  of a Taylor series as
• f(xi ?x) f(xi) (df/dx)xi (?x) (d2f/dx2)xi
  (?x2)/ 2! .. (dnf/dxn)xi (?xn)/n! .
• The above series converges if ?x is small and
  f(x) is differentiable
• For a converging series, successive terms are
  progressively smaller

20
Finite Difference Methods
The terms in the Taylor series expansion can be
rearranged to give (df/dx)xi f(xi ?x) -f(xi)
/?x-(d2f/dx2)xi (?x)/2! --(dnf/dxn)xi (?xn-1)/n!
-... Or (2) (df/dx)xi f(xi ?x) -f(xi)
/?xO(?x) Here O(?x) implies that the leading
term in the neglected terms are of the order of
?x, i.e., the error in the approximation reduces
by a factor of 2 if ?x is halved. Equation (2)
is therefore a first order-accurate approximation
for the first derivative.
21
Finite Difference Methods
Other Approximations for a First
Derivative Other approximations are also
possible. Writing the Taylor series expansion for
f(xi- ?x), we have (3) f(xi-?x) f(xi)
(df/dx)xi (?x) (d2f/dx2)xi (?x2)/ 2! -..
(dnf/dxn)xi (?xn)/n! Equation (3) can be
rearranged to give another first order
approximation (4) (df/dx)xi f(xi) -f(xi-?x)
/?xO(?x) Subtracting (3) from (1) gives a
second order approximation for df/dx (5)
(df/dx)xi f(xi ?x) -f(xi-?x) /(2?x) O(?x2)
22
Finite Difference Methods
In summary Forward-difference formula (first
order-accurate approximation) (df/dx)xi f(xi
?x) -f(xi) /?xO(?x) Backward-difference
formula (first order-accurate approximation) (df
/dx)xi f(xi) -f(xi-?x) /?xO(?x) Central-dif
ference formula (second order-accurate
approximation) (df/dx)xi f(xi ?x) -f(xi-?x)
/(2?x) O(?x2)
23
Finite Difference Methods
Geometrical interpretation of finite differences
Given a function f(x) shown in Fig. 2, we can
approximate its derivative, slope or tangent at P
by the slope of the arcs PB, PA, or AB, for
obtaining the forward difference,
backward-difference, and central-difference
formulas respectively. In general, a second
order correct analog is always better
approximation than is a first order correct analog
24
Finite Difference Methods
Higher derivates Upon adding (1) and (3) (1)
f(xi ?x) f(xi) (df/dx)xi (?x) (d2f/dx2)xi
(?x2)/ 2! .. (dnf/dxn)xi (?xn)/n! . (3)
f(xi-?x) f(xi) (df/dx)xi (?x) (d2f/dx2)xi
(?x2)/ 2! -.. (dnf/dxn)xi (?xn)/n! we
obtain f(xi ?x) - f(xi-?x) 2f(xi)
(d2f/dx2)xi (?x2)/ 2! O(?x4) and we
have (d2f/dx2)xi f(xi ?x) - 2f(xi)
f(xi-?x) /(?x2) O(?x2) (second order-accurate
approximation) Higher order finite difference
approximations can be obtained by taking more
terms in Taylor series expansion.
25
Type of Errors
1)Discretization or truncation error
eiyi-y(xi) The discretization error encountered
in integrating a differential equation across one
step is sometimes called local truncation
error. This error is determined solely by the
particular numerical solution procedure selected,
that is, by the nature of the approximations
present in the method this type of error is
independent of computing equipment
characteristics. Example when one replace the
first derivate by the finite Forward-difference
formula, the truncation error is of the order Dx,
while it is of the order Dx2 when a
Central-difference formula is used.
26
Finite Difference Methods
Example Forward-difference formula (the
truncation error is of the order of
Dx) (df/dx)xi f(xi ?x) -f(xi)
/?xO(?x) Backward-difference formula (the
truncation error is of the order of
Dx) (df/dx)xi f(xi) -f(xi-?x)
/?xO(?x) Central-difference formula (the
truncation error is of the order of
Dx2) (df/dx)xi f(xi ?x) -f(xi-?x) /(2?x)
O(?x2)
27
Type of Errors
2)The round-off error is determined, for any
particular method, by the computing
characteristics of the machine which does the
calculations due to its finite memory which lead
to a an approximation of any irrational number by
a rounded value.
28
Type of Errors
Reducing the mesh size could increase accuracy,
but the mesh size could not be infinitesimal.
Decreasing the truncation error by using a
finer mesh may result in increasing the round-off
error due to the increased number of arithmetic
operations. A point is reached where minimum
total error occurs for any particular algorithm
using any given word length.
29
Finite Difference Methods
Analog finite difference equation for each total
derivates Forward-difference formula (first
order-accurate approximation) (df/dx)xi f(xi
?x) -f(xi) /?xO(?x) Backward-difference
formula (first order-accurate approximation) (df
/dx)xi f(xi) -f(xi-?x) /?xO(?x) Central-dif
ference formula (second order-accurate
approximation) (df/dx)xi f(xi ?x) -f(xi-?x)
/(2?x) O(?x2) Second derivate (second
order-accurate approximation) (d2f/dx2)xi
f(xi ?x) - 2f(xi) f(xi-?x) /(?x2) O(?x2)
30
Finite Difference analog to the partial derivates
Partial derivatives of u at the (i,n)th node
Forward
Backward
Central
Forward
31
Finite Difference scheme
Estimating derivatives numerically
Finite difference equations (algebraic equations)
32
Finite Differencing of Parabolic PDEs
Consider a simple example of a parabolic (or
diffusion) partial differential equation with one
spatial independent variable with a
(diffusion coefficient) constant.
x?xA,xB I.C. u(x,0)uo(x) tgt0 B.C.1
u(0,t)uA(t) B.C.2 u(xN,t)uB(t)
33
Finite Differencing of Parabolic PDEs
To apply the difference method to find the
solution of a function u(x,t), we divide the
solution region in x-t plane into equal
rectangles or meshes of sides ?x and ?t. x
i?x, i1,2,,N t n?t . n1,2,
34
Finite Differencing of Parabolic PDEs
In the development of the analog finite
difference equation we write
All i and n
Both sides must to be evaluated at the same
conditions xi and tn.
Then we substitute the analog finite difference
equation for each derivates.
35
Finite Difference analog to the partial derivates
Partial derivatives of u at the (i,n)th node
Forward
Backward
Central
Forward
36
Finite Difference scheme

• Depending on analog finite difference equation
  used one can obtain different finite difference
  equations analogous to the
• Eulers Method (explicit or forward)
• Laasonens Method (implicit o backward)
• Crank-Nicholson Method (implicit)

37
Finite Difference scheme
Eulers Method (explicit or forward)
The forward or explicit difference equation is
probably the most well known, although it is the
least efficient of all the possible equations
which can be used. We use the forward difference
formula for the derivative with respective to t
and central difference formula with respect to x.
38
Finite Difference scheme
Eulers Method (explicit or forward)
Forward
Central
39
Finite Difference scheme
Eulers Method (explicit or forward)
Finite difference equation
i2N-1
This equation contains only one unknown value,
uin1, and is written explicitly for this
unknown. Therefore, this explicit formula can be
used to compute u(xi,tn1) explicitly in terms of
u(xi,tn) for i2..N-1 (internal grid points).
The computation of the values of the dependent
variable is thus made one point at a time.
40
Finite Difference scheme
Eulers Method (explicit or forward)
Finite difference equation
I.C.
i2N-1
The values of u along the first time row (n0 or
t0) can be calculated in terms of the boundary
and initial conditions, then the values of u
along the second row (n1 or t?t) are calculated
in terms of the first time row, and so on
41
Finite Difference scheme
Eulers Method (explicit or forward)
The explicit formula is simple to implement, and
especially easy to use for computing the value of
u at each time level, but the its computation is
slow. In general, a numerical solution must
converge to the solution of the corresponding
differential equation when the finite increments
, Dx and Dt, are decreased in size. Analysis as
shown that a very restrictive relationship
between the size of Dx and that of Dt must be
satisfied in order for the solution of forward
equation to approach that of the differential
equation. The restriction required that One
should use very small Dt and large Dx which
results to a stable solution but with low
accuracy (in order to minimize the truncation
error in the x analogs (O (Dx2)), the size of Dx
must be small).
42
Finite Difference scheme
Eulers Method (explicit or forward)
The many transient problems which have boundary
conditions independent of time approach a steady
state condition. For these problems The
time increment can be continuously increased and
made quite large as the solution progresses
towards steady state without causing significant
truncation error in the time analog. However, for
the forward difference equation, the size of Dt
must remain on the order of (Dx)2 for the
solution to be stable. Thus, a very small value
of Dt must be used for stability even when larger
value could be used without causing truncation
error. A finite difference equation which does
not have this restriction is, therefore, a much
better one to use as an analog to the
differential equation
x?xA,xBI.C. u(x,0)uo(x) tgt0 B.C.1
u(0,t)uA B.C.2 u(xN,t)uB
43
Finite Difference scheme
Laasonens Method (implicit o backward)
In searching for a new finite difference
equation which does not have a restriction on the
size of Dt for stability, we might write the
finite differences analogs for the derivates at
the new or unknown time which is indexed by n1.
44
Finite Difference scheme
Laasonens Method (implicit o backward)
Note The backward one is the only possible to be
implement
45
Finite Difference scheme
Laasonens Method (implicit o backward)
46
Finite Difference scheme
Laasonens Method (implicit o backward)
(1)
i2N-1
This equation is implicit that is, it contains
three values of the dependent values , u, at the
unknown time level, n1. For N-1 increments in x
and the boundary conditions of equation B.C.1
u(0,t)uA B.C.2 u(xN,t)uB there will be
N-2 unknown values of u at each time level, and
there are N-2 equations, one corresponding to
each points. In summary N-2 equations (1) for
2ltiltN-1 B.C.1 for i1 u1n1uA B.C.2 for iN
uNn1uB
47
Finite Difference scheme
Laasonens Method (implicit o backward)
The resulting set of equation represents a linear
system in which the coefficient matrix is
tridiagonal. If N7, for each time level, one
should calculate uin1 as B.C1
u1n1uA B.C.2 u7n1uB The
equation can thus be solved by the Thomas method
to obtain N-2 values of uin1. The same method
can be used to obtain the values uin2 from the
now known values of uin1
48
Finite Difference scheme
Laasonens Method (implicit o backward)

• There is no restriction on the size of Dt for
  stability of the backward difference equation.
• The size of Dt can therefore be set,
  independently of the size of Dx, to minimize the
  truncation error of the Taylor series in time.
• All the values of u at the first unknown time
  level (n1) will be computed from the initial
  values (n0).
• In the solution of these equations on a digital
  computer, it is usual practice to keep the size
  of Dx constant throughout the calculation.

49
Finite Difference scheme
Laasonens Method (implicit o backward)
If the boundary conditions are of the form of
equation B.C.1 u(0,t)uA B.C.2
u(xN,t)uB so that the transient solution
approaches a steady state condition, it is usual
practice to increase the size of Dt as the
solution progresses in order to obtain the
solution in a minimum computing time. This is
because as the steady-state conditions are
approached, the difference in values of u from
one time level to the next diminishes if a
constant time increment is used. Similarly, as
the steady-state conditions are approached, the
size of second and the higher time derivates
decrease thus, the same truncation error will
result from a larger Dt as the steady-state is
approached.
50
Finite Difference scheme
Laasonens Method (implicit o backward)

• The backward difference equation is an efficient
  one, and it is simple to use.
• However, it is only first-order correct in time.
• It is desirable to find a second-order correct
  analog to this derivates

first order-accurate approximation
second order-accurate approximation
51
Finite Difference scheme
Crank-Nicholson Method (implicit)
In the Crank-Nicholson equation all the finite
difference are written about the point (xi,
tn1/2) (point designated by a cross) which is
halfway between the known and the unknown time
levels. Values of the dependent variable, u,
are computed only at the points designated by
circles.
52
Finite Difference scheme
Crank-Nicholson Method (implicit)
Second order correct analog of the time derivate
at the point (xi, tn1/2) (Central formula at
the time level tn1/2)
Truncation error
53
Finite Difference scheme
Crank-Nicholson Method (implicit)
The second order space derivate at the point (xi,
tn1/2)
n1/2
The second order space derivate at the point (xi,
tn1/2) is approximated by the arithmetic
average of the central difference formulas at the
point (xi, tn) and (xi, tn1)
54
Finite Difference scheme
Crank-Nicholson Method (implicit)
Crank-Nicholson finite difference equation which
is second-order correct in both x and t
55
Finite Difference scheme
Crank-Nicholson Method (implicit)
This equation is implicit as it contains the same
three unknown values of u that are found in the
backward difference equation. Similarly, this
equation and its two boundary conditions can be
solved by the Thomas method.
56
Finite Difference scheme
Crank-Nicholson Method (implicit)

• The Crank-Nicholson Method requires more
  computations per time step than the backward
  difference method to evaluate the right side of
  the equation (this side contains values of u at
  three length positions at the known time level
  instead of the one value found in the backward
  difference equation.
• However, a larger time increment can be used for
  the Crank-Nicholson equation , since its time
  derivate analog is second order correct, fewer
  time steps are thus necessary to computes values
  of the dependent variable for a given elapsed
  time.
• Thus the Crank-Nicholson equation is more
  efficient than the backward difference method and
  is the preferred method for obtaining numerical
  solutions to parabolic differential equations.

57
Finite Difference scheme
Crank-Nicholson Method (implicit)
The Crank-Nicholson equation, like the backward
difference equation, is stable for all ratios of
Dx to Dt.
58
Finite Difference scheme
Crank-Nicholson Method (implicit)
In matrix form the system of equations becomes
Vector of unknown values
Vector of known values
59
Finite Difference scheme
Crank-Nicholson Method (implicit)
The resulting set of equation represents a linear
system in which the coefficient matrix is
tridiagonal. If N7, for each time level, one
should calculate uin1 as B.C1
u1n1uA B.C.2 u7n1uB The
equation can thus be solved by the Thomas method
to obtain N-2 values of uin1. The same method
can be used to obtain the values uin2 from the
now known values of uin1
60
Numerical methods to solve parabolic
PDEs Boundary conditions
61
Boundary conditions
2 order PDE 2 independent variables Linear with
constant coefficients ? is a diffusion coefficient
I.C. t 0 u(x,0) u0(x)
The boundary conditions for partial differential
equations of second order must to be two and they
can arise as expressions containing the values
??of the unknown function u and / or its first
derivatives ?u/?x
62
Boundary conditions
Dirichlet condition
The condition of the Dirichlet is the simplest
case of boundary conditions. In general, it
provides the boundary conditions for the possible
dependence on the time variable through the
function C (t) and F (t) (which can also be
constant) We write the finite difference
equation at the internal nodes of the domain
including the nodes adjacent to the boundary
(i2N-1). Then in this equations we replace the
value of the function on the boundary value
(known) imposed by the condition.
B.C.1 x0 -gt u(t, x0)C(t) all t (simple
case) B.C.2 xL -gt u(t, xL)F(t) all t
(simple case)
63
Boundary conditions
Dirichlet condition
Example
I.C. t 0 u(x,0) u0(x)
B.C.1 x0 -gt u(t, x0)C(t) all t (simple
case) B.C.2 xL -gt u(t, xL)F(t) all t
(simple case) where C(t) and F(t) can be also
constant
64
Boundary conditions
Dirichlet condition
Example Laasonens Method (implicit o
backward)
(1)
i2N-1
B.C1 x0 u1n1uA(t) B.C.2 xL uNn1uB(t)
Neq(N-2) (1) 2BC N
65
Boundary conditions
Neumann condition
The case of the Neumann condition differs from
the previous by the fact that the unknown
function is also unknown on the boundary, on
which, however, the derivative is known.
Therefore, for the same discrete grid, you must
obtain one equation more (for each Neumann
condition) than in the Dirichlet case.
66
Boundary conditions
Neumann condition
Example
I.C. t 0 u(x,0) u0(x)
B.C.1 x0 -gt u(t, x0)C(t) all t
(simple case) B.C.2 xL -gt u(t, xL)
-gt(?u/?x)q all t (simple case) where q is
known
The finite-difference equation must be write, as
in the previous case, for all internal nodes
(i2..N-1), and the boundary condition will
provide an additional equation for the unknown
value of the function on the boundary
67
Boundary conditions
Neumann condition (1)
Example Laasonens Method
(1)
B.C1 x0 u1n1uA(t) B.C.2 xL
Backward formula
Neq(N-2) (1) 2BC N
68
Boundary conditions
Neumann condition

B.C.2 xL
Backward formula
Note The expression used for the first
derivative, being "backward" and not central, is
only accurate to first order. To improve this
aspect, it is useful to write a formula that is
central to the position of the edge (iN), which
is when we know the value of the first
derivative. This node can be obtained by
considering an outside "fictitious node".
69
Boundary conditions
Neumann condition (2)
Using the following discretization one can
write the discretized equation also for the N
node as where uN1n1 is the value taken
by function in the fictitious node

1 2 N-2
N-1 N N1
x0
xL
i2..N -gt N-1 equations
70
Boundary conditions
Neumann condition (2)
In summary

0 1 N-2
N-1 N N1
x0
xL
i2..N -gt N-1 equations
BC2 xL ? (Central formula)
71
Boundary conditions
Neumann condition
In summary
Dx
Dx

0 1 N-2
N-1 N N1
x0
xL
Again we have N equations in N unknowns, and thus
all of the formulas employed for the derivatives
of space are the second order. It should also
be noted that the formula used for the first
derivative is formally second order accurate, but
for a double-interval (2Dx).
72
Boundary conditions
Neumann condition


A further improvement can be achieved using the
following discretization Note the grid
is distributed so as to drop the last pair of
nodes, formed by the last node (N) and the
outside fictitious node (N1), straddling the
boundary on which the Neumann condition is
imposed. In this way we write the balance
equation, as in the previous case, for all
internal nodes to the node N, while the boundary
condition will be written as
Dx/2
Dx/2
BC2 xL ?
(Central formula)
73
Boundary conditions
Neumann condition (3)


In summary
i2..N -gt N-1 equations
Central and therefore accurate to second order,
and for a range of Dx.
BC2 xL ?
The latter method is thus preferable to others
when the Neumann condition is "pure, that is,
not involving the boundary value of the function.
74
Boundary conditions
Third type or mixed condition


The flow is not assigned as a value (q), but is
determined by an exchange coefficient multiplied
by a driving force which is a function of the
unknown value taken by the function on the edge,
thus giving rise to a boundary condition known as
third type or mixed. It will be more
natural adopt the second method, which involves
both the fictitious node and the node at the edge.
Example ?
75
Convergence and stability of FDM for 2 order
linear parabolic Equations
Definition of convergence


U(x,t) is the exact solution of partial
differential problem in IR 't. u is the exact
solution of finite difference equations used to
approximate the differential problem.
Intuitively, it is expected to be a "good"
method when u-gtU when Dx, Dt -gt0 More
precisely, one can say for example that, if uin
is the value of u calculated at the point (xi,
tn) of the integration domain, and if then
the finite difference method is said to be
convergent. If U is the vector generated by
evaluating U(xi, tn), then U-u is said to be the
discretization error, and is the direct
consequence of the truncation error in finite
difference formulas.
76
Convergence and stability of FDM for 2 order
linear parabolic Equations
Definition of numerical stability

• u is the exact solution of finite difference
  equations used to approximate the differential
  problem
• u is the numerical solution of the same problem
• As a result of rounding errors (roundoff)
  introduced by the machine is always
• u?u
• A method is said to be stable if (u u) does not
  diverges as the number of discretization nodes
  increases.
• Eulers method is stable when
  (Stability Parameter)
• Both Laasonens Method and Crank-Nicholson
  equations, are stable for all ratios of Dx to Dt.
  
• (Stability Parameter)

77
Convergence and stability of FDM for 2 order
linear parabolic Equations
Definition of consistency


A finite difference scheme is called consistent
if, making the limit for finite increments of the
independent variables that tend to zero, it
returns the differential expression that you want
to approximate. Note all the three schemes
examined in this course are consistent. Of
course, what matters is the construction of
methods which can prove the convergence, since at
the end the only solution is becoming available
is the numerical one which, hopefully, is close
to the exact one. Convergence Laxs theorem A
finite difference method for a second order
linear PDE schemes that use consistent scheme,
and that is numerically stable, is convergent.
78
Parabolic PDEs
Parabolic (or diffusion) PDE with one spatial
independent variable general form
t ? 0, 8 x ? xA, xB I.C. u(t0,x)u(x)
all x B.C.1 u(t, xxA)uA(t) all t (simple
case) B.C.2 u(t, xxB)uB(t) all t (simple
case) Analytical solution uu(t,x) where u is a
continuous function of t and x
79
Parabolic PDEs
PARABOLIC PDE Linear generation term
Explicit method