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An approximation of a differential equation and its boundary conditions by a system of finite (usually algebraic) equations giving the values of the unknown function on some grid, which is subsequently made more exact by making the parameter of the finite-difference problem (the step of the grid, the mesh) tend to zero.
 
An approximation of a differential equation and its boundary conditions by a system of finite (usually algebraic) equations giving the values of the unknown function on some grid, which is subsequently made more exact by making the parameter of the finite-difference problem (the step of the grid, the mesh) tend to zero.
  
Consider the computation of a function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129201.png" /> which belongs to a linear normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129202.png" /> of functions defined in a given domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129203.png" /> with boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129204.png" />, and which is the solution of the differential boundary value problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129205.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129206.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129207.png" /> is a differential equation, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129208.png" /> is the set of boundary conditions. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a0129209.png" /> be a grid (cf. [[Approximation of a differential operator by difference operators|Approximation of a differential operator by difference operators]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292010.png" /> be the normed linear space of functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292011.png" /> defined on this grid. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292012.png" /> be a table of values of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292013.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292014.png" />. A norm is introduced into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292015.png" /> so that the equality
+
Consider the computation of a function $  u $
 +
which belongs to a linear normed space $  U $
 +
of functions defined in a given domain $  D _{U} $
 +
with boundary $  \Gamma $,  
 +
and which is the solution of the differential boundary value problem $  Lu = 0 $,  
 +
$  lu \mid  _ \Gamma  = 0 $,  
 +
where $  Lu = 0 $
 +
is a differential equation, while $  lu \mid  _ \Gamma  = 0 $
 +
is the set of boundary conditions. Let $  D _{ {hU}} $
 +
be a grid (cf. [[Approximation of a differential operator by difference operators|Approximation of a differential operator by difference operators]]) and let $  U _{h} $
 +
be the normed linear space of functions $  u _{h} $
 +
defined on this grid. Let $  [v] _{h} $
 +
be a table of values of the function $  v $
 +
at the points of $  D _{ {hU}} $.  
 +
A norm is introduced into $  U _{h} $
 +
so that the equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292016.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {h \rightarrow 0} \  \| [ v ] _{h} \| _{ {U _ h}} \  = \  \| v \| _{U}  $$
  
is valid for any function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292017.png" />. The problem of computing the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292018.png" /> is replaced by a certain problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292019.png" /> for the approximate computation of the table <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292020.png" /> of values of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292021.png" /> at the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292022.png" />. Here, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292023.png" /> is a certain set of (non-differential) equations for the values of the grid function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292024.png" />.
+
is valid for any function $  v \in U $.  
 +
The problem of computing the solution $  u $
 +
is replaced by a certain problem $  {\mathcal L} _{h} u _{h} = 0 $
 +
for the approximate computation of the table $  [u] _{h} $
 +
of values of $  u $
 +
at the points of $  D _{ {hU}} $.  
 +
Here, $  {\mathcal L} _{h} u _{h} $
 +
is a certain set of (non-differential) equations for the values of the grid function $  u _{h} \in U _{h} $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292025.png" /> be an arbitrary function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292026.png" />, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292027.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292028.png" /> be the normed linear space to which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292029.png" /> belongs for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292030.png" />. One says that the problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292031.png" /> is a finite-difference approximation of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292032.png" /> of the differential boundary value problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292034.png" />, on the space of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292035.png" /> of the latter if
+
Let $  v _{h} $
 +
be an arbitrary function of $  U _{h} $,  
 +
let $  {\mathcal L} _{h} v _{h} = \phi _{h} $,  
 +
and let $  \Phi _{h} $
 +
be the normed linear space to which $  \phi _{h} $
 +
belongs for any $  v _{h} \in U _{h} $.  
 +
One says that the problem $  {\mathcal L} _{h} u _{h} $
 +
is a finite-difference approximation of order $  p $
 +
of the differential boundary value problem $  Lu = 0 $,  
 +
$  lu \mid  _ \Gamma  =0 $,  
 +
on the space of solutions $  u $
 +
of the latter if
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292036.png" /></td> </tr></table>
+
$$
 +
\| {\mathcal L} _{h} [ u ] _{h} \| _ { \Phi _ h } \  = \
 +
O ( h^{p} ) .
 +
$$
  
The actual construction of the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292037.png" /> involves a separate construction of its two subsystems <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292039.png" />. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292040.png" /> one uses the difference approximations of a differential equation (cf. [[Approximation of a differential equation by difference equations|Approximation of a differential equation by difference equations]]). The complementary equations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292041.png" /> are constructed using the boundary conditions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292042.png" />.
+
The actual construction of the system $  {\mathcal L} _{h} u _{h} $
 +
involves a separate construction of its two subsystems $  L _{h} u _{h} = 0 $
 +
and $  l _{h} u _{h} \mid  _{ {\Gamma _ h}} = 0 $.  
 +
For $  L _{h} u _{h} = 0 $
 +
one uses the difference approximations of a differential equation (cf. [[Approximation of a differential equation by difference equations|Approximation of a differential equation by difference equations]]). The complementary equations $  l _{h} u _{h} \mid  _{ {\Gamma _ h}} = 0 $
 +
are constructed using the boundary conditions $  lu _{h} \mid  _ \Gamma  = 0 $.
  
An approximation such as has just been described never ensures [[#References|[2]]] that the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292043.png" /> of the finite-difference problem converges to the exact solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292044.png" />, i.e. that the equality
+
An approximation such as has just been described never ensures [[#References|[2]]] that the solution $  u _{h} $
 +
of the finite-difference problem converges to the exact solution $  u $,  
 +
i.e. that the equality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292045.png" /></td> </tr></table>
+
$$
 +
\lim\limits _ {h \rightarrow 0} \  \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \  = 0
 +
$$
  
is valid, no matter how the norms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292047.png" /> have been chosen.
+
is valid, no matter how the norms in $  U _{h} $
 +
and $  \Phi _{h} $
 +
have been chosen.
  
The additional condition, the fulfillment of which in fact ensures convergence, is stability [[#References|[3]]], [[#References|[5]]]–[[#References|[8]]], which must be displayed by the finite-difference problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292048.png" />. This problem is called stable if there exist numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292049.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292050.png" /> such that the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292051.png" /> has a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292052.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292053.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292054.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292055.png" />, and if this solution satisfies the inequality
+
The additional condition, the fulfillment of which in fact ensures convergence, is stability [[#References|[3]]], [[#References|[5]]]–[[#References|[8]]], which must be displayed by the finite-difference problem $  {\mathcal L} _{h} u _{h} = 0 $.  
 +
This problem is called stable if there exist numbers $  \delta > 0 $
 +
and $  0 < h _{0} $
 +
such that the equation $  {\mathcal L} _{h} z _{h} = \phi _{h} $
 +
has a unique solution $  z _{h} \in U _{h} $
 +
for any $  \phi _{h} \in \Phi _{h} $,  
 +
$  \| \phi _{h} \| < \delta $,  
 +
$  h < h _{0} $,  
 +
and if this solution satisfies the inequality
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292056.png" /></td> </tr></table>
+
$$
 +
\| z _{h} -u _{h} \| _{ {U _ h}} \  \leq \  C \  \| \phi _{h} \| _{ {\Phi _ h}} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292057.png" /> is a constant not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292058.png" /> or on the perturbation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292059.png" /> of the right-hand side, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292060.png" /> is a solution of the unperturbed problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292061.png" />. If a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292062.png" /> of the differential problem exists, while the finite-difference problem <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292063.png" /> approximates the differential problem on solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292064.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292065.png" /> and is stable, then one has convergence of the same order, i.e.
+
where $  C $
 +
is a constant not depending on $  h $
 +
or on the perturbation $  \phi _{h} $
 +
of the right-hand side, while $  u _{h} $
 +
is a solution of the unperturbed problem $  {\mathcal L} _{h} u _{h} = 0 $.  
 +
If a solution $  u $
 +
of the differential problem exists, while the finite-difference problem $  {\mathcal L} _{h} u _{h} $
 +
approximates the differential problem on solutions $  u $
 +
of order $  p $
 +
and is stable, then one has convergence of the same order, i.e.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292066.png" /></td> </tr></table>
+
$$
 +
\| [ u ] _{h} -u _{h} \| _{ {U _ h}} \  = \  O ( h^{p} ) .
 +
$$
  
 
For instance, the problem
 
For instance, the problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292067.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
$$ \tag{1}
 +
\left . { {
 +
L (u) \  \equiv \ 
 +
\frac{\partial u}{\partial t}
 +
-  
 +
\frac{\partial u}{\partial x}
 +
= \
 +
0,\ \  t > 0,\ \  - \infty < x < \infty ,} \atop {
 +
| lu | _ \Gamma  \  = \  u ( 0 ,\  x ) - \psi (x) \  = \  0,\ \  - \infty
 +
< x < \infty ,
 +
}} \right \}
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292068.png" /> is a given function with a bounded second-order derivative, can be approximated, for a natural definition of the norms, by the finite-difference problem
+
where $  \psi (x) $
 +
is a given function with a bounded second-order derivative, can be approximated, for a natural definition of the norms, by the finite-difference problem
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292069.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
$$ \tag{2}
 +
{\mathcal L} _{h} u _{h} \  = \
 +
\left \{ { {
 +
L _{h} u _{h} \  =
 +
\frac{u _ m^{n+1} -u _{m} ^ n} \tau
 +
-
 +
\frac{u _ m+1^{n} -u _{m} ^ n}{h}
 +
= \  0} \atop {| l _{h} u _{h} | _ { \Gamma _ h } \  \equiv \
 +
u _ m^{0} - \psi (mh) \  = \  0}}
 +
\right \}\  =
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292070.png" /></td> </tr></table>
+
$$
 +
= \
 +
0 \  \in \  \Phi _{h} ,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292071.png" /> is the value of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292072.png" /> at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292074.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292075.png" />. If the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292076.png" /> is taken to be the upper bound of the moduli of the right-hand sides of the equations which constitute the system <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292077.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292078.png" />, then the approximation of problem (1) by problem (2) on solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292079.png" /> is of the first order. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292080.png" />, there is no convergence, whatever the norm. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292081.png" /> and the norm is
+
where $  u _ m^{n} $
 +
is the value of $  u _{h} $
 +
at $  ( x _{m} ,\  t _{n} ) = (mh,\  n \tau ) $,  
 +
$  \tau = rh $,  
 +
$  r = \textrm{ const } $.  
 +
If the norm of $  \phi _{n} $
 +
is taken to be the upper bound of the moduli of the right-hand sides of the equations which constitute the system $  {\mathcal L} _{h} v _{h} = \phi _{h} $,  
 +
$  v _{h} \in U _{h} $,  
 +
then the approximation of problem (1) by problem (2) on solutions $  u $
 +
is of the first order. If $  r > 1 $,  
 +
there is no convergence, whatever the norm. If $  r \leq 1 $
 +
and the norm is
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292082.png" /></td> </tr></table>
+
$$
 +
\| u _{h} \| _{ {U _ h}} \  = \  \sup _ { m,n } \  | u _ m^{n} | ,
 +
$$
  
 
the problem is stable, so that there is convergence [[#References|[2]]], [[#References|[3]]]:
 
the problem is stable, so that there is convergence [[#References|[2]]], [[#References|[3]]]:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292083.png" /></td> </tr></table>
+
$$
 +
\| [ u ] _{h} -u _{h} \| _{ {U _ h}} \  = \  O (h) .
 +
$$
  
 
The replacement of differential problems by difference problems is one of the most universal methods for the approximate computation of solutions of differential boundary value problems on a computer [[#References|[7]]].
 
The replacement of differential problems by difference problems is one of the most universal methods for the approximate computation of solutions of differential boundary value problems on a computer [[#References|[7]]].
  
The replacement of differential problems by their difference analogues started in the works [[#References|[1]]], [[#References|[2]]] and [[#References|[4]]], and is sometimes employed to prove that the differential problem is in fact solvable. This is done as follows. It is proved that the set of solutions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292084.png" /> of the difference analogue of the differential boundary value problem is compact with respect to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292085.png" />, after which a proof is given that a solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292086.png" /> of the differential boundary value problem is the limit of a subsequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292087.png" /> which converges as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292088.png" />. If this solution is known to be unique, then not only the subsequence, but also the entire set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292089.png" /> converges to the solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292090.png" /> as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/a/a012/a012920/a01292091.png" />.
+
The replacement of differential problems by their difference analogues started in the works [[#References|[1]]], [[#References|[2]]] and [[#References|[4]]], and is sometimes employed to prove that the differential problem is in fact solvable. This is done as follows. It is proved that the set of solutions $  u _{h} $
 +
of the difference analogue of the differential boundary value problem is compact with respect to $  h $,  
 +
after which a proof is given that a solution $  u $
 +
of the differential boundary value problem is the limit of a subsequence $  u _{ {h _ k}} $
 +
which converges as $  h _{k} \rightarrow 0 $.  
 +
If this solution is known to be unique, then not only the subsequence, but also the entire set of $  u _{h} $
 +
converges to the solution $  u $
 +
as $  h \rightarrow 0 $.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  "Dirichlet's problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1940)  pp. 125–124  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,  K. Friedrichs,  H. Lewy,  "Ueber die partiellen Differenzengleichungen der mathematischen Physik"  ''Math. Ann.'' , '''100'''  (1928)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "New existence proofs for the solution of the Dirichlet problem by the method of finite differences"  ''Uspekhi Mat. Nauk'' , '''8'''  (1940)  pp. 161–170  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.S. Ryaben'kii,  "On the application of the method of finite differences to the solution of the Cauchy problem"  ''Dokl. Akad. Nauk SSSR'' , '''86''' :  6  (1952)  pp. 1071–1073  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. [V.S. Ryaben'kii] Rjabenki,  A.F. [A.F. Filippov] Filipov,  "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Samarskii,  "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Filippov,  "On stability of difference equations"  ''Dokl. Akad. Nauk SSSR'' , '''100''' :  6  (1955)  pp. 1045–1048  (In Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  L.A. Lyusternik,  "Dirichlet's problem"  ''Uspekhi Mat. Nauk'' , '''8'''  (1940)  pp. 125–124  (In Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  R. Courant,  K. Friedrichs,  H. Lewy,  "Ueber die partiellen Differenzengleichungen der mathematischen Physik"  ''Math. Ann.'' , '''100'''  (1928)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.K. Godunov,  V.S. Ryaben'kii,  "The theory of difference schemes" , North-Holland  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  I.G. Petrovskii,  "New existence proofs for the solution of the Dirichlet problem by the method of finite differences"  ''Uspekhi Mat. Nauk'' , '''8'''  (1940)  pp. 161–170  (In Russian)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  V.S. Ryaben'kii,  "On the application of the method of finite differences to the solution of the Cauchy problem"  ''Dokl. Akad. Nauk SSSR'' , '''86''' :  6  (1952)  pp. 1071–1073  (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  V.S. [V.S. Ryaben'kii] Rjabenki,  A.F. [A.F. Filippov] Filipov,  "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft.  (1960)  (Translated from Russian)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  A.A. Samarskii,  "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D.  (1984)  (Translated from Russian)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  A.F. Filippov,  "On stability of difference equations"  ''Dokl. Akad. Nauk SSSR'' , '''100''' :  6  (1955)  pp. 1045–1048  (In Russian)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
For additional references, see the additional references to [[Approximation of a differential operator by difference operators|Approximation of a differential operator by difference operators]].
 
For additional references, see the additional references to [[Approximation of a differential operator by difference operators|Approximation of a differential operator by difference operators]].

Latest revision as of 20:44, 29 January 2020


An approximation of a differential equation and its boundary conditions by a system of finite (usually algebraic) equations giving the values of the unknown function on some grid, which is subsequently made more exact by making the parameter of the finite-difference problem (the step of the grid, the mesh) tend to zero.

Consider the computation of a function $ u $ which belongs to a linear normed space $ U $ of functions defined in a given domain $ D _{U} $ with boundary $ \Gamma $, and which is the solution of the differential boundary value problem $ Lu = 0 $, $ lu \mid _ \Gamma = 0 $, where $ Lu = 0 $ is a differential equation, while $ lu \mid _ \Gamma = 0 $ is the set of boundary conditions. Let $ D _{ {hU}} $ be a grid (cf. Approximation of a differential operator by difference operators) and let $ U _{h} $ be the normed linear space of functions $ u _{h} $ defined on this grid. Let $ [v] _{h} $ be a table of values of the function $ v $ at the points of $ D _{ {hU}} $. A norm is introduced into $ U _{h} $ so that the equality

$$ \lim\limits _ {h \rightarrow 0} \ \| [ v ] _{h} \| _{ {U _ h}} \ = \ \| v \| _{U} $$

is valid for any function $ v \in U $. The problem of computing the solution $ u $ is replaced by a certain problem $ {\mathcal L} _{h} u _{h} = 0 $ for the approximate computation of the table $ [u] _{h} $ of values of $ u $ at the points of $ D _{ {hU}} $. Here, $ {\mathcal L} _{h} u _{h} $ is a certain set of (non-differential) equations for the values of the grid function $ u _{h} \in U _{h} $.

Let $ v _{h} $ be an arbitrary function of $ U _{h} $, let $ {\mathcal L} _{h} v _{h} = \phi _{h} $, and let $ \Phi _{h} $ be the normed linear space to which $ \phi _{h} $ belongs for any $ v _{h} \in U _{h} $. One says that the problem $ {\mathcal L} _{h} u _{h} $ is a finite-difference approximation of order $ p $ of the differential boundary value problem $ Lu = 0 $, $ lu \mid _ \Gamma =0 $, on the space of solutions $ u $ of the latter if

$$ \| {\mathcal L} _{h} [ u ] _{h} \| _ { \Phi _ h } \ = \ O ( h^{p} ) . $$

The actual construction of the system $ {\mathcal L} _{h} u _{h} $ involves a separate construction of its two subsystems $ L _{h} u _{h} = 0 $ and $ l _{h} u _{h} \mid _{ {\Gamma _ h}} = 0 $. For $ L _{h} u _{h} = 0 $ one uses the difference approximations of a differential equation (cf. Approximation of a differential equation by difference equations). The complementary equations $ l _{h} u _{h} \mid _{ {\Gamma _ h}} = 0 $ are constructed using the boundary conditions $ lu _{h} \mid _ \Gamma = 0 $.

An approximation such as has just been described never ensures [2] that the solution $ u _{h} $ of the finite-difference problem converges to the exact solution $ u $, i.e. that the equality

$$ \lim\limits _ {h \rightarrow 0} \ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ 0 $$

is valid, no matter how the norms in $ U _{h} $ and $ \Phi _{h} $ have been chosen.

The additional condition, the fulfillment of which in fact ensures convergence, is stability [3], [5][8], which must be displayed by the finite-difference problem $ {\mathcal L} _{h} u _{h} = 0 $. This problem is called stable if there exist numbers $ \delta > 0 $ and $ 0 < h _{0} $ such that the equation $ {\mathcal L} _{h} z _{h} = \phi _{h} $ has a unique solution $ z _{h} \in U _{h} $ for any $ \phi _{h} \in \Phi _{h} $, $ \| \phi _{h} \| < \delta $, $ h < h _{0} $, and if this solution satisfies the inequality

$$ \| z _{h} -u _{h} \| _{ {U _ h}} \ \leq \ C \ \| \phi _{h} \| _{ {\Phi _ h}} , $$

where $ C $ is a constant not depending on $ h $ or on the perturbation $ \phi _{h} $ of the right-hand side, while $ u _{h} $ is a solution of the unperturbed problem $ {\mathcal L} _{h} u _{h} = 0 $. If a solution $ u $ of the differential problem exists, while the finite-difference problem $ {\mathcal L} _{h} u _{h} $ approximates the differential problem on solutions $ u $ of order $ p $ and is stable, then one has convergence of the same order, i.e.

$$ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ O ( h^{p} ) . $$

For instance, the problem

$$ \tag{1} \left . { { L (u) \ \equiv \ \frac{\partial u}{\partial t} - \frac{\partial u}{\partial x} \ = \ 0,\ \ t > 0,\ \ - \infty < x < \infty ,} \atop { | lu | _ \Gamma \ = \ u ( 0 ,\ x ) - \psi (x) \ = \ 0,\ \ - \infty < x < \infty , }} \right \} $$

where $ \psi (x) $ is a given function with a bounded second-order derivative, can be approximated, for a natural definition of the norms, by the finite-difference problem

$$ \tag{2} {\mathcal L} _{h} u _{h} \ = \ \left \{ { { L _{h} u _{h} \ = \ \frac{u _ m^{n+1} -u _{m} ^ n} \tau - \frac{u _ m+1^{n} -u _{m} ^ n}{h} \ = \ 0} \atop {| l _{h} u _{h} | _ { \Gamma _ h } \ \equiv \ u _ m^{0} - \psi (mh) \ = \ 0}} \right \}\ = $$

$$ = \ 0 \ \in \ \Phi _{h} , $$

where $ u _ m^{n} $ is the value of $ u _{h} $ at $ ( x _{m} ,\ t _{n} ) = (mh,\ n \tau ) $, $ \tau = rh $, $ r = \textrm{ const } $. If the norm of $ \phi _{n} $ is taken to be the upper bound of the moduli of the right-hand sides of the equations which constitute the system $ {\mathcal L} _{h} v _{h} = \phi _{h} $, $ v _{h} \in U _{h} $, then the approximation of problem (1) by problem (2) on solutions $ u $ is of the first order. If $ r > 1 $, there is no convergence, whatever the norm. If $ r \leq 1 $ and the norm is

$$ \| u _{h} \| _{ {U _ h}} \ = \ \sup _ { m,n } \ | u _ m^{n} | , $$

the problem is stable, so that there is convergence [2], [3]:

$$ \| [ u ] _{h} -u _{h} \| _{ {U _ h}} \ = \ O (h) . $$

The replacement of differential problems by difference problems is one of the most universal methods for the approximate computation of solutions of differential boundary value problems on a computer [7].

The replacement of differential problems by their difference analogues started in the works [1], [2] and [4], and is sometimes employed to prove that the differential problem is in fact solvable. This is done as follows. It is proved that the set of solutions $ u _{h} $ of the difference analogue of the differential boundary value problem is compact with respect to $ h $, after which a proof is given that a solution $ u $ of the differential boundary value problem is the limit of a subsequence $ u _{ {h _ k}} $ which converges as $ h _{k} \rightarrow 0 $. If this solution is known to be unique, then not only the subsequence, but also the entire set of $ u _{h} $ converges to the solution $ u $ as $ h \rightarrow 0 $.

References

[1] L.A. Lyusternik, "Dirichlet's problem" Uspekhi Mat. Nauk , 8 (1940) pp. 125–124 (In Russian)
[2] R. Courant, K. Friedrichs, H. Lewy, "Ueber die partiellen Differenzengleichungen der mathematischen Physik" Math. Ann. , 100 (1928)
[3] S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)
[4] I.G. Petrovskii, "New existence proofs for the solution of the Dirichlet problem by the method of finite differences" Uspekhi Mat. Nauk , 8 (1940) pp. 161–170 (In Russian)
[5] V.S. Ryaben'kii, "On the application of the method of finite differences to the solution of the Cauchy problem" Dokl. Akad. Nauk SSSR , 86 : 6 (1952) pp. 1071–1073 (In Russian)
[6] V.S. [V.S. Ryaben'kii] Rjabenki, A.F. [A.F. Filippov] Filipov, "Über die stabilität von Differenzgleichungen" , Deutsch. Verlag Wissenschaft. (1960) (Translated from Russian)
[7] A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[8] A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian)

Comments

For additional references, see the additional references to Approximation of a differential operator by difference operators.

How to Cite This Entry:
Approximation of a differential boundary value problem by difference boundary value problems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Approximation_of_a_differential_boundary_value_problem_by_difference_boundary_value_problems&oldid=44368
This article was adapted from an original article by V.S. Ryaben'kii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article