Approximation of a differential equation by difference equations

An approximation of a differential equation by a system of algebraic equations for the values of the unknown functions on some grid, which is made more exact by making the parameter (mesh, step) of the grid tend to zero.

Let $ L, Lu = f $, be some differential operator and let $ L _ {h} , L _ {h} u _ {h} = f _ {h} ,\ u _ {h} \in U _ {h} , f _ {h} \in F _ {h} $, be some finite-difference operator (cf. Approximation of a differential operator by difference operators). One says that the finite-difference expression $ L _ {h} u _ {h} = 0 $, $ 0 \in F _ {h} $, is an approximation of order $ p $ to the differential equation $ Lu=0 $ on solutions $ u $ if the operator $ L _ {h} $ approximates the operator $ L $ on solutions $ u $ of order $ p $, i.e. if

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

The simplest example of the construction of a finite-difference equation $ L _ {h} u _ {n} = 0 $ which approximates the differential equation $ Lu = 0 $ on solutions $ u $, consists in replacing each derivative in the expression $ Lu $ by its finite-difference analogue.

For instance, the equation

$$ Lu \equiv \frac{d ^ {2} u }{d x ^ {2} } + p (x) \frac{du}{dx} + q (x) u = 0 $$

is approximated with second-order accuracy by the finite-difference equation

$$ L _ {h} u _ {h} \equiv \frac{u _ {m+1} -2 u _ {m} + u _ {m-1} }{h} ^ {2} + $$

$$ + p ( x _ {m} ) \frac{u _ {m+1} -u _ {m-1} }{2h} + q ( x _ {m} ) u _ {m} = 0, $$

where the grids $ D _ {hU} $ and $ D _ {hF} $ consist of points $ x _ {m} = mh $, $ m $ is an integer, and $ u _ {m} $ is the value of the function $ u _ {h} $ at $ x _ {m} $. Again, the equation

$$ Lu \equiv \frac{\partial u }{\partial t } - \frac{\partial ^ {2} u }{\partial x ^ {2} } = 0, $$

is approximated by two different difference approximations on smooth solutions:

$$ L _ {h} ^ {(1)} u _ {h} \equiv \ \frac{u _ {n} ^ {n+1} -u _ {m} ^ {n} } \tau - \frac{u _ {m+1} ^ {n} -2 u _ {m} ^ {n} +u _ {m-1} ^ {n} }{h} ^ {2} = 0 $$

(explicit scheme), and

$$ L _ {h} ^ {(2)} u _ {h} \equiv \ \frac{u _ {m} ^ {n+1} -u _ {m} ^ {n} } \tau - \frac{u _ {m+1} ^ {n+1} -2u _ {m} ^ {n+1} + u _ {m-1} ^ {n+1} }{h} ^ {2} = 0 $$

(implicit scheme), in which the grids $ D _ {hU} $ and $ D _ {hF} $ consist of the points $ (x _ {m} , t _ {n} ) = (mh, n \tau ) $, $ \tau = r h ^ {2} $, $ r = \textrm{ const } $, $ m $ and $ n $ are integers, and $ u _ {m} ^ {n} $ is the value of the function $ u _ {h} $ at the point $ ( x _ {m} , t _ {n} ) $ of the grid. There exist finite-difference operators $ L _ {h} $ which represent an approximation to the differential operator $ L $ especially well only on solutions $ u $ of the equation $ Lu = 0 $ and less well on other functions. For instance, the operator $ L _ {h} $,

$$ L _ {h} u _ {n} \equiv \ \frac{u _ {m+1} -u _ {m} }{h} - \phi \left ( x _ {m} + \frac{h}{2} , \frac{u _ {m} + \widetilde{u} }{2} \right ) = \ f _ {h} \left ( x _ {m} + \frac{h}{2} \right ) , $$

where $ \widetilde{u} = u _ {m} + h \phi (x _ {m} ) $, is a first-order approximation (with respect to $ h $) of the operator $ L $,

$$ Lu \equiv \frac{du}{dx} - \phi ( x , u ) = \ f ( x ) , $$

on arbitrary smooth functions $ u(x) $ and it is a second-order approximation on solutions of the equation $ Lu = 0 $( it is assumed that the function $ u $ is sufficiently smooth). When finding numerical solutions of boundary value problems for the differential equation $ Lu = 0 $ with the aid of the finite-difference equation $ L _ {h} u _ {h} = 0 $, it is the approximation property of $ L _ {h} $ when applied to solutions $ u $ of $ Lu = 0 $ that is significant and not its approximation property when applied to arbitrary smooth functions. For a large class of differential equations and systems of equations there exist methods for constructing difference equations approximating them, which also meet various additional conditions: stability of the solution $ u _ {h} $ with respect to rounding-off errors such as are permitted in the calculations; the validity of certain integral relations for $ u _ {h} $ which hold for the solution $ u $ of the differential equation; the permissibility of using arbitrary grids $ D _ {hU } $ and $ D _ {hF } $( this is important in calculating the motion of a continuous medium); a limitation on the number of arithmetical operations required to compute the solutions; etc.

The approximation of a differential equation by difference equations is an element of the approximation of a differential boundary value problem by difference boundary value problems in order to approximately calculate a solution of the former.

References

[1]	S.K. Godunov, V.S. Ryaben'kii, "The theory of difference schemes" , North-Holland (1964) (Translated from Russian)
[2]	A.A. Samarskii, "Theorie der Differenzverfahren" , Akad. Verlagsgesell. Geest u. Portig K.-D. (1984) (Translated from Russian)
[3]	S.K. Godunov, et al., "Numerical solution of multi-dimensional problems of gas dynamics" , Moscow (1976) (In Russian)
[4]	A.A. Samarskii, Yu.P. Popov, "Difference schemes of gas dynamics" , Moscow (1975) (In Russian)

Comments

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