Finite Difference Scheme for Special System of Partial Differential Equations

Abstract

The paper establishes conditions of existence and uniqueness of the bounded solution of a special system of linear partial differential equations of the first order. The system arises in the problem of a finite difference scheme of finding an approximate solution is elaborated.

1 Problem Statement

Further $E^n$ is the  Euclidean space of vectors x, (T denotes the transposition) with the norm $\Vert x\Vert$; Z is the set of integers; $Z^n$ is the n-dimensional Cartesian product. We consider a problem of numerical solving of finding on $[0,T] \times E^n$ of a system of partial differential equations

$$\begin{aligned} \frac{\partial l^{(k)(t, x)}}{\partial t} + \sum _{i=1}^n f^{(i)} (t, x) \frac{\partial l^{(k)(t, x)}}{\partial x} + \sum _{i=1}^n g_k^{(i)} (t, x) l^{(i)} (t, x) = q^{(k)} (t, x),\quad k = \overline{1, n}. \end{aligned}$$

(1)

With initial conditions

$$\begin{aligned} l^{(k)} (0, x) = r^{(k)},\quad k = \overline{1, n}. \end{aligned}$$

(2)

Further we assume that the following hypotheses be fulfilled.

Assumption 1

The problem (1)–(2) has continuous differentiable on $[0,T] \times E^n$ solution $l(t, x) \left( l^{(1)}(t, x), \ldots , l^{(n)} (t, x) \right)$ such that partial derivatives $\frac{\partial ^2 l^{(k)} (t, x)}{\partial t^2}$ and $\frac{\partial ^2 l^{(k)} (t, x)}{\partial {x_i}^2}, i = 1, \ldots , n, k = 1, \ldots , n$ are continuous and bounded on $[0,T] \times E^n$. Also we assume existence of constants F, G such that

$$\begin{aligned} \Vert f(t, x) \Vert \le F(t, x) \in [0, T]; \end{aligned}$$

(3)

$$\begin{aligned} g_k (t, x) \le G(t, x) \in [0, T] \times E^n,\quad k = \overline{1, n}. \end{aligned}$$

(4)

2 Finite Difference Scheme: Approximation, Stability, Convergence

Let $\alpha = (\alpha _1, \ldots , \alpha _n) \in E^n ; \bar{e_k}$ be the unite vector of the axis $0x_k$ and $\tau = T/M$ (M is a natural). Denote $t_\nu = \nu \tau$; $\nu = 0, \ldots , M$; $x^\alpha = \alpha _1 h \bar{e_1}, \ldots , \alpha _n h \bar{e_n}$; $f_{\nu , \alpha } = f(t_\nu , x^\alpha )$.

In the region $[0, T]\times E^n$ we construct grids $\varOmega ^0 _h = {(0, x^\alpha ): \alpha \in Z^n}, \varOmega ^ \nu _h = {(t_ \nu , x^ \alpha ): \nu = 0, \ldots , M}$; $\varOmega ^{'} _h = \left\{ (t_\nu , x^\alpha ): \nu = 1, \ldots , M; \alpha \in Z^n \right\} .$

For grid functions $u_{\nu , \alpha } = \left( u^{(1)} _{\nu , \alpha }, \ldots , u^{(n)} _{\nu , \alpha } \right)$ defined on grids $\varOmega ^{\nu }_h$ and $\varOmega ^{'}_h$ we use the corresponding norms

$$ u_{\nu , \alpha } = \sup {\varOmega _h} \Vert u_{\nu , \alpha } \Vert , u_{\nu , \alpha } = \sup {\varOmega ^{'}_h} \Vert u_{\nu , \alpha } \Vert . $$

Let $n^{+}_{\nu , \alpha } = {j \in {1, \ldots , n}: f ^{(j)} _{\nu , \alpha } > 0}, n^{-}_{\nu , \alpha } = {j \in {1, \ldots , n}: f ^{(j)} _{\nu , \alpha } \le 0}.$

The difference numerical scheme corresponding to the problem (1)–(2) we construct in the following way.

On the grid $\varOmega ^{'}_{h}$:

$$\begin{aligned} \begin{aligned}&\frac{u^{(k)}_{\nu , \alpha } - u^{(k)}_{(\nu -1), \alpha }}{\tau } + \sum _{i \in n^{+}_{\nu , \alpha }} f^{(i)}_{(\nu -1), \alpha } \frac{u^{(k)}_{(\nu -1), \alpha } - u^{(k)}_{(\nu -1), \alpha - \bar{e_l}}}{h} + \\&+ \sum _{i \in n^{-}_{\nu , \alpha }} f^{(i)}_{(\nu -1), \alpha } \frac{u^{(k)}_{(\nu -1), \alpha } - u^{(k)}_{(\nu -1), \alpha }}{h} + \sum _{i=1}^n g^{(i)}_{k, (\nu =1), \alpha } u^{(i)}_{(\nu -1), \alpha } = q^{(k)}_{(\nu -1), \alpha },\quad k = \overline{1, n}. \end{aligned} \end{aligned}$$

(5)

On the grid $\varOmega ^0_h$:

$$\begin{aligned} u^{(k)}_{0, \alpha } = r^{(k)}_\alpha , \quad k = \overline{1, n}. \end{aligned}$$

(6)

From (5)

$$\begin{aligned} \begin{aligned} u^{(k)}_{\nu , \alpha }&= \left( 1 - \frac{\tau }{h} \sum _{i=1}^n \left| f^{(i)}_{\nu -1, \alpha } \right| \right) u^{(k)}_{\nu -1, \alpha } +\\&+ \frac{\tau }{h} \sum _{i \in n^+_{\nu -1, \alpha }} f^{(i)}_{\nu -1, \alpha } \times u^{(k)}_{\nu -1, \alpha - \bar{e_l}} - \frac{\tau }{h} \sum _{i \in n^-_{\nu -1, \alpha }} f^{(i)}_{\nu -1, \alpha } \times u^{(k)}_{\nu -1, \alpha + \bar{e_l}}. \end{aligned} \end{aligned}$$

Solving the Eq. (5) with respect to $u_{\nu , \alpha }$ obtain

$$\begin{aligned} \begin{aligned} u_{\nu , \alpha }&= \left( 1 - \frac{\tau }{h} \sum _{i=1}^n f^{(i)}_{\nu -1, \alpha } \right) f^{(k)}_{\nu -1, \alpha } + \frac{\tau }{h} \sum _{i \in n} f^{(i)}_{\nu -1, \alpha } \times u^{(k)}_{\nu -1, \alpha - \bar{e_l}} - \frac{\tau }{h} + \\&+ \frac{\tau }{h} \sum _{i \in n_{\nu -1, \alpha }} f^{(i)}_{\nu -1, \alpha } \times u^{(k)}_{\nu -1, \alpha + \bar{e_l}} - \tau \sum _{i=1}^n g^{(i)}_{k, \nu -1, \alpha } u^{(i)}_{\nu -1, \alpha } + \tau q^{(k)}_{\nu -1, \alpha } , k = \overline{1, n} \end{aligned} \end{aligned}$$

(7)

Because $u^{(k)}_{0, \alpha }$ are known from the initial condition (6) then by the formula (7) one can calculate layer by layer at first $u_{1, \alpha }, \alpha \in Z_n$, then $u_{2 \alpha }, \alpha \in Z_n$, and so on.

Let us estimate the approximation order which the scheme (5)–(6) approximates the problem (1)–(2). Due to the Assumption 1 according to the Taylor series we have

$$\begin{aligned} \frac{l^{(k)}(t_\nu , x^\alpha ) - l^{(k)}(t_{\nu -1}, x^\alpha )}{\tau } = \frac{\partial l^{(k)}(t_{\nu -1}, x^\alpha )}{\partial t} + \frac{\tau }{2} \frac{\partial ^2 l^{(k)}(t_\nu , x^\alpha )}{\partial t^2} \end{aligned}$$

(8)

$$ \frac{l^{(k)}(t_{\nu -1}, x^\alpha ) - l^{(k)}(t_{\nu -1}, x^\alpha - h\bar{e_l})}{h} = \frac{\partial l^{(k)}(t_{\nu -1}, x^\alpha )}{\partial x_i} - $$

$$\begin{aligned} - \frac{h}{2} \frac{\partial ^2 l^{(k)}(t_\nu -1, \xi ^{k, \nu , \alpha })}{\partial {x_i}^2}, \quad i = \overline{1, n}, \end{aligned}$$

(9)

$$\begin{aligned} \frac{h}{2} \frac{\partial ^2 l^{(k)}(t_\nu -1, \eta ^{k, \nu , \alpha }_i)}{\partial {x_i}^2}, \quad i = \overline{1, n} \end{aligned}$$

(10)

where

$$\begin{aligned} t_\nu \le \xi ^{k}_{\nu , \alpha } \le t_ {\nu , x^\alpha - h\bar{e_{l}}} \le \xi _{i}^{k, \nu , \alpha } \le x^{\alpha }, \quad x^{\alpha } \le \eta _{i}^{k, \nu , \alpha } \le x^{\alpha } +h\bar{e_l}. \end{aligned}$$

(11)

From (6) follows that the initial condition (2) is approximated at $\varOmega _h^0$ exactly. Then due to (9)–(11) the residual between (1) and (5) on the solution l(t, x) is equal to

$$\begin{aligned} \begin{aligned} \delta ^{(k)}_{t, h}&= \frac{\tau }{2} \frac{\partial ^2 l^{(k)} \left( \xi ^{(k)}_{\nu , \alpha } , x^\alpha \right) }{\partial {t^2}} - \frac{h}{2} \sum ^+_{i \in n_{\nu -1, \alpha }} f^{(i)}_{\nu -1, \alpha } \frac{\partial ^2 l^{(k)} \left( t_{\nu -1, \alpha } \xi ^{(k)}_{\nu , \alpha }, x_{\alpha } \right) }{\partial {t^2}} \\&+ \frac{h}{2} \sum ^-_{i \in n_{\nu -1, \alpha }} f^{(i)}_{\nu -1, \alpha } \frac{\partial ^2 l^{(k)} \left( t_{\nu -1, \alpha } \eta ^{(k, \nu , \alpha )}_{i} \right) }{\partial {x_i ^2}},\quad k=\overline{1, n} \end{aligned} \end{aligned}$$

Due to the Assumption 1 the estimation $\Vert \delta \Vert \le c \times (\tau +h)$, $c = const$ is valid from which follows the following proposition.

Theorem 1

If the Assumption 1 is valid then the difference scheme (5)–(6) approximates the problem (1)–(2) on its solution l(t, x) with the first order with respect to $\tau$ and h.

Let us show the stability of the difference scheme (5)–(6). It will be sufficiently for its convergence, because the initial condition (2) is approximated exactly on $\varOmega ^0_h$.

Formula (7) shows the solvability of the difference problem (5)–(6). Let us obtain estimation of the solution of (5) corresponding to the zero initial conditions

$$\begin{aligned} u^{(k)}_{0, \alpha } = 0,\quad k = \overline{1, n}. \end{aligned}$$

(12)

If

$$\begin{aligned} 0 < \frac{\tau }{h} \le \frac{1}{nF}, \end{aligned}$$

(13)

then from (3), (4), (7) follows

$$ \sup _\alpha \Vert u_{\nu , \alpha } \Vert \le (1+ \tau Gn) \sup _\alpha \Vert u_{\nu -1, \alpha } \Vert + \tau {\Vert q_{\nu , \alpha }\Vert }^{'}_h. $$

Then taking into account (12), obtain

$$ \sup _{\nu , \alpha } \Vert u_{\nu , \alpha } \Vert \le \frac{T}{M} {\Vert q_{\nu , \alpha }\Vert }^{'}_h \left( 1+ \frac{TGn}{M} \right) ^M \times $$

$$\begin{aligned} \times \left[ \frac{1}{(1+ \tau GM)^M} + \frac{1}{(1+ \tau GM)^{M-1}} + \cdots + \frac{1}{1 + \tau GM} \right] . \end{aligned}$$

(14)

Taking into account that $(1 + \frac{TGn}{M})^M$ tends to $e^{TGn}$ as $M \rightarrow \infty$ and therefore is bounded, then from (14) follows that the solution $u_{\nu , \alpha }$ of the problem (5), (12) satisfies the estimation $\left\| u_{\nu , \alpha } \right\| _h \le L \left\| q_{\nu , \alpha } \right\| _h, L = const.$ This proves the following proposition.

Theorem 2

If conditions (3), (4) and (8) are fulfilled then the scheme (5)–(6) is stable with respect to the right-hand side. From the stability and the approximation of the difference scheme follows its convergence.

Theorem 3

Let the Assumption 1 and conditions (3), (4) and (8) be fulfilled then the solution of the difference scheme (5)–(6) converges to the solution of the problem (1)–(2) with the first order by $\tau$ and h.