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Unformatted text preview: F or implicit RungeKutta software that control accuracy through automatic time step and order variation. Implicit RungeKutta methods may be derived
as nite element methods by using the Galerkin method (9.3.1) with higherorder trial
and test functions. Of the many possibilities, we'll examine a class of methods where the
trial function c(t) is discontinuous.
Example 9.3.2. Suppose that c(t) is a polynomial on (tn tn+1 ] with jump discontinuities at tn, n 0. When we need to distinguish left and right limits, we'll use the
notation cn; = lim c(tn ; )
!0 cn+ = lim c(tn + ):
!0 (9.3.4a) With jumps at tn, we'll have to be more precise about the temporal inner product (9.3.1b)
and we'll de ne
(u v)n; = lim
!0 Zt n+1 tn ; ; uvdt (u v)n+ = lim
!0 Zt n+1 tn + ; uvdt: (9.3.4b) The inner product (u v)n; may be a ected by discontinuities in functions at tn, but
(u v)n+ only involves integrals of smooth functions. In particular:
(u v)n; = (u v)n+ when u(t) and v(t) are either continuous or have jump discontinuities at tn
(u v)n; exists and (u v)n+ = 0 when either u or v are proportional to the delta
function (t ; tn) and
(u v)n; doesn't exist while (v u)n+ = 0 when both u and v are proportional to
(t ; tn).
Suppose, for example, that v(t) is continuous at tn and u(t) = (t ; tn). Then
(u v)n; = lim
!0 Zt n+1 tn ; ; (t ; tn )v(t)dt = v(tn): The delta function can be approximated by a smooth function that depends on as was
done in Section 3.2 to help explain this result.
Let us assume that w(t) is continuous and write c(t) in the form c(t) = cn; + c(t) ; cn;]H (t ; tn) (9.3.5a) 16 Parabolic Problems where t>0
H (t) = 1 ioftherwise
0
is the Heaviside function and c is a polynomial in t. (9.3.5b) Di erentiating _
_
c(t) = c(t) ; cn;] (t ; tn ) + c(t)H (t ; tn): (9.3.5c) With the interpretation that inner products in (9.3.1) are of type (9.3.4), assume that
w(t) is continuous and use (9.3.5) in (9.3.1a) to obtain
_
wT (tn)M(tn)(cn+ ; cn;) + (w Mc)n+ + (w Kc)n+ = (w l)n+ 8w 2 H 1: (9.3.6) The simplest discontinuous Galerkin method uses a piecewise constant (p = 0) basis
in time. Such approximations are obtained from (9.3.5a) by selecting c(t) = cn+ = c(n+1);:
Testing against the constant function w(t) = 1 1 : : : 1]T
and assuming that M and K are independent of t, (9.3.6) becomes M(c (n+1); ; c ) + Kc
n; (n+1); t= Zt n+1 tn l(t)dt: The result is almost the same as the backward Euler formula (9.2.11b) except that the
load vector l is averaged over the time step instead of being evaluated at tn+1.
With a linear (p = 1) approximation for c(t), we have c(t) = cn+Nn(t) + c(n+1);Nn+1 (t)
where Nn+i, i = 0 1, are given by (9.3.2b). Selecting the basis for the test space as wi(t) = Nn+i(t) 1 1 : : : 1]T i=0 1 assuming that M and K are independent of t, and substituting the above approximations
into (9.3.6), we obtain
1 M(c(n+1); ; cn+) + t K(2cn+ + c(n+1); ) = Z tn+1 N l(t)dt
M(c ; c ) + 2
n
6
tn
n+ n; 9.3. Finite Element Methods in Time 17 1 M(c(n+1); ; cn+) + t K(cn+ + 2c(n+1); ) = Z tn+1 N l(t)dt:
n+1
2
6
tn
Simplifying the expressions and assuming that l(t) can be approximated by a linear
function on (tn tn+1) yields the system and n+...
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 Spring '14
 JosephE.Flaherty
 The Land

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