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Unformatted text preview: F or implicit Runge-Kutta software that control accuracy through automatic time step and order variation. Implicit Runge-Kutta methods may be derived
as nite element methods by using the Galerkin method (9.3.1) with higher-order 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
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+ 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:
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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