Approximation of a differential operator by difference operators

An approximation of the differential operator by parameter-dependent operators such that the result of their application to a function is determined by the values of this function on some discrete set of points — a grid — which become more exact as its parameter (mesh, step of the grid) tends to zero.

Let $L, Lu = f$, be a differential operator which converts any function $u$ of a class of functions $U$ into a function $f$ of a linear normed space $F$. Let $D _ {U}$ be the domain of definition of the functions in $U$, and let there be some discrete subset in $D _ {U}$ — a grid $D _ {hU }$ — which "becomes more dense" as $h \rightarrow 0$. Consider the set $U _ {h}$ of all functions $[u] _ {h}$ defined on the grid only and coinciding with $u$ in the points of the grid. A difference operator is defined as any operator $L _ {h}$ that converts the grid functions in $U _ {h}$ into functions $f _ {h}$ in $F$. One says that the operator $L _ {h}$, $L _ {h} [u] _ {h} = f _ {h}$, represents an order $p$ approximation to the differential operator $L$ on $U$ if for any function $u \in U$

$$\| Lu - L _ {h} [ u ] _ {h} \| _ {F} \rightarrow 0 ,$$

$$\| Lu - L _ {h} [ u ] _ {h} \| _ {F } \leq ch ^ {p} ,\ c = c ( u ) = \textrm{ const } ,$$

as $h \rightarrow 0$. Occasionally, an approximation is understood to be the equality

$$\lim\limits _ {h \rightarrow 0 } \ L _ {h} [ u ] _ {h} = Lu$$

in the sense of some weak convergence. The approximation of a differential operator by difference operators is used for an approximate computation of the function $Lu$ from the table of values $[u] _ {h}$ of the function $u$ and for the approximation of a differential equation by difference equations.

There are two principal methods for constructing operators $L _ {h}$ approximating $L$.

In the first method $L _ {h} [u] _ {h}$ is defined as the result of applying the differential operator $L$ to a function in $U$, obtained by some interpolation formula from the grid function $[u] _ {h}$.

The second method is as follows. In the domain $D _ {F}$ of definition of a function $f$ in $F$ one introduces a grid $D _ {hF }$, and considers the linear space $F _ {h}$ of grid functions defined on $D _ {hF}$. The operator $L _ {h} [u] _ {h}$ is constructed as the product of two operators: an operator which converts the function $[u] _ {h}$ into the grid function $f _ {h}$ in $F _ {h}$, i.e. into a table of approximate values of $f$, and an operator which extends $f _ {h}$ from $D _ {hF }$ to the entire domain $D _ {f}$. For instance, in order to approximate the differential operator

$$L = \frac{d}{dx} ,\ \frac{du}{dx} = f (x),\ \ 0 \leq z \leq 1 ,$$

one constructs the grid $D _ {hU }$ consisting of points $x _ {k}$, $k = 0 \dots N$,

$$0 = x _ {0} < \dots < x _ {k} < x _ {k+1} < \dots < x _ {N} = 1 ,$$

$$\max _ { k } ( x _ {k + 1 } - x _ {k} ) = h ,$$

and a grid $D _ {hF }$ consisting of the points

$$x _ {k} ^ {*} = x _ {k} + \theta ( x _ {k + 1 } - x _ {k} ) ,\ k = 0 \dots N - 1 ,$$

$$0 \leq \theta \leq 1,\ \theta = \textrm{ const } .$$

The values of the operator $L _ {h} [u] _ {h}$ at the points $x _ {k} ^ {*}$ are defined by the equations:

$$\left . L _ {h} [ u ] _ {h} \right | _ {x = x _ {k} ^ {*} } = \ f _ {h} ( x _ {k} ^ {*} ) = \frac{u ( x _ {k+1} ) -u ( x _ {k} ) }{x _ {k+1} - x _ {k} } ,$$

$$k = 0 \dots N - 1 .$$

Thereafter, the definition of $L _ {h} [u] _ {h}$ is piecewise linearly extended outside $D _ {hF }$ with possible breaks at the points $x _ {k} ^ {*}$, $k = 1 \dots N - 2$, only.

Let the norm in F be defined by the formula

$$\| \phi \| _ {F} = \sup _ { x } | \phi (x) | .$$

Then, on the class of functions $U$ with a bounded third derivative, for $\theta = 0$ and $\theta = h/2$ the operator $L _ {h}$ represents an order 1, respectively 2, approximation to $L = d/dx$. On the class $U$ of functions with bounded second derivatives, the representation is of order 1 only, for any $\theta \in [0, 1]$.

The task of approximating a differential operator by finite-difference operators is sometimes conditionally considered as solved if a method is found for the construction of the grid function

$$\left . L _ {h} [ u ] _ {h} \right | _ {D _ {hF} } = f _ {h} \in F _ {h} ,$$

determined at the points of $D _ {hF }$ only, while the task of completing the function $f _ {h}$ everywhere on $D _ {F}$ is ignored. In such a case the approximation is defined by considering the space $F _ {h}$ as normed, and by assuming, for the grid and for the norm, that for any function $f \in F$, the function $\{ f \} _ {h} \in F _ {h}$, which coincides with $f$ at the points of $D _ {hF }$, satisfies the equation

$$\lim\limits _ {h \rightarrow 0 } \| \{ f \} _ {h} \| _ {F _ {h} } = \| f \| _ {F} .$$

The operator $L _ {h}$ is understood to be an operator from $U _ {h}$ in $F _ {h}$, and one says that $L _ {h}$ represents an order $p$ approximation to $L$ on $U$ if, for $h \rightarrow 0$,

$$\| \{ Lu \} _ {h} - L _ {h} [ u ] _ {h} \| _ {F _ {h} } \rightarrow 0 ,$$

$$\| \{ Lu \} _ {h} - L _ {h} [ u ] _ {h} \| _ {F _ {h} } \leq ch ^ {p} .$$

In order to construct an operator $L _ {h}$ which is an approximation to $L$ of given order on sufficiently smooth functions, one often replaces each derivative contained in the expression $L$ by its finite-difference approximation, basing oneself on the following fact. For any integers $i, j$ and for any $k _ {0}$, $2 k _ {0} + 1 \geq i + j$, in the equation

$$h ^ {-j} \sum _ { k = - k _ {0} } ^ {k _ 0 } c _ {k} u ( x + kh ) = u ^ {(j)} ( x ) + \epsilon ( x , h , c _ {-k _ {0} } \dots c _ {k _ {0} } ),$$

it is possible, by using the method of undetermined coefficients and Taylor's formula, to select numbers $c _ {k}$ not depending on $h$, so that for any function $u(x)$ with $j + r$( $r \leq i$) bounded derivatives, an inequality of the type

$$| \epsilon ( x , h , c _ {- k _ {0} } \dots c _ {k _ {0} } ) | \leq A _ {ij} \sup _ { t } \ | u ^ {( j + r ) } ( t ) | h ^ {r} ,$$

where $A _ {ij }$ depends only on $i$ and $j$, is valid. As an example, suppose one constructs an approximating operator for the Laplace operator $\Delta$,

$$\Delta u \equiv \frac{\partial ^ {2} u }{\partial x ^ {2} } + \frac{\partial ^ {2} u }{\partial y ^ {2} } = f ( x , y ) ,$$

if $D _ {U}$ is the closed square $| x | \leq 1, | y | \leq 1$, and $D _ {F}$ is its interior $| x | < 1, | y | < 1$. Assume that $h = 1/N$ where $N$ is a natural number, and construct the grid with points

$$( x , y ) = ( mh, nh ) ,\ \ | mh | \leq 1,\ | nh | \leq 1,$$

which belong to $D _ {hU }$. The points

$$( x , y ) = ( mh , nh ) ,\ \ | mh | \leq 1,\ | nh | \leq 1,$$

then belong to $D _ {hF }$, for integers $m$ and $n$. Since

$$\frac{1}{h ^ {2} } [ y ( x + h ) -2 y ( x ) + y ( x - h ) ] = y ^ {\prime\prime} ( x ) + h ^ \frac{2}{12} y ^ {(4)} ( \xi ) ,$$

$\Delta$ can be approximated with second-order accuracy on a space of sufficiently smooth functions by the finite-difference operator $L _ {h}$ if one puts, at the points of $D _ {hF }$:

$$L _ {h} u _ {m,n} = \frac{u _ {m+1,n} -2u _ {m,n} +u _ {m-1,n} }{h ^ {2} } +$$

$$+ \frac{u _ {m,m+1} -2u _ {m,n} +u _ {m,n-1} }{h ^ {2} } = f _ {mn} ,$$

where $u _ {m,n }$ and $f _ {m,n }$ are the values of the functions $[u] _ {h}$ and $f _ {h}$ at the point $(mh, nh)$.

There are also other methods of constructing operators $L _ {h}$ which are approximations to the operator $L$ on the space of solutions $u$ of the differential equation $Lu = 0$, and which satisfy additional conditions.

References

 [1] A.F. Filippov, "On stability of difference equations" Dokl. Akad. Nauk SSSR , 100 : 6 (1955) pp. 1045–1048 (In Russian) [2] I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian)