Space Discretization

We next use the variational statement derived previously to make a discretization of the space dimension. Therefore we introduce a partition $\mathcal{T}_h : jh$, $j = 0,1,\ldots, N$ of the interval $0 \leq x \leq 1$ into $N$ subintervals of equal length $h = 1/N$ and let $V_h$ be the corresponding space of continuous piecewise linear functions vanishing at $x=0$. We now seek a solution approximation $U(x,t) \in V_h$ such that for every fixed time $t$

$$\int_0^1 \dot U v dx + \int_0^1 U' v' dx = 0,$$ (8)

for all $v \in V_h$.

Further, if $\{ \varphi_j \}_1^N$ denotes a set of nodal basis functions of $V_h$ we can expand $U(x)$ as

$$U(x) = \xi_1(t) \varphi_1(x) + \xi_2(t) \varphi_2(x) + \ldots + \xi_N(t) \varphi_N(x) = \sum_{j=1}^N \xi_j(t) \varphi_j(x).$$ (9)

Notice that the coefficients $\xi_j(t)$ are time dependent but not space dependent functions.

Substituting $U(x,t)$ into (7) and choosing test functions $v = \varphi_j$, $j = 1,2,\ldots, N$ we obtain

$$\left \{ \begin{array}{c} \displaystyle \int_0^1 (\dot \xi_1(t) \... ...'_2 + \ldots \xi_N(t)\varphi'_N) \varphi'_N dx = 0 \end{array} \right . ,$$ (10)

which is a $(N \times N)$ system of ordinary differential equations (ODE)

$$\mathchoice{\hbox{\boldmath$\displaystyle M$}} {\hbox{\boldmath$\... ...}} {\hbox{\boldmath$\scriptstyle 0$}} {\hbox{\boldmath$\scriptscriptstyle 0$}},$$ (11)

where the entries of the matrices $\mathchoice{\hbox{\boldmath $\displaystyle M$}} {\hbox{\boldmath $\textstyle M$}} {\hbox{\boldmath $\scriptstyle M$}} {\hbox{\boldmath $\scriptscriptstyle M$}}$ and $\mathchoice{\hbox{\boldmath $\displaystyle A$}} {\hbox{\boldmath $\textstyle A$}} {\hbox{\boldmath $\scriptstyle A$}} {\hbox{\boldmath $\scriptscriptstyle A$}}$ are given by

$$M_{ij} = \int^1_0 \varphi_i(x) \varphi_j(x) dx, \qquad A_{ij} = \int^1_0 \varphi'_i(x) \varphi'_j(x) dx, \qquad i,j = 1,2,\ldots, N.$$ (12)

Here $\mathchoice{\hbox{\boldmath $\displaystyle \xi$}} {\hbox{\boldmath $\textstyle... ...x{\boldmath $\scriptstyle \xi$}} {\hbox{\boldmath $\scriptscriptstyle \xi$}}(t)$ denotes a vector holding the nodal values $\xi_i(t)$ of $U(x,t)$.