Adaptivity with moving grids

54 C. J. Budd, W. Huang and R. D. Russell
where φ j , (dφ j /ds) and (d 2 φ j /d 2 s), j=1,...,4, are functions of s (i) .These
expressions for U and its derivatives can then be directly substituted into
(3.15), and the expression
U t = f(t, x, U, U x , U xx )
evaluated at the two Gauss points X ij = X i + s j H i ,j =1, 2, where
s 1 = 1 (
1 − 1 )
√ , s 2 = 1 (
1+ √ 1 ).
2 3 2 3
This, when coupled to the boundary conditions of the underlying PDE,
leads to a set of ordinary differential equations for U i , U i,x which can then
be coupled directly to the ODEs for the moving mesh given, for example,
by (3.11). In certain circumstances, such as when the underlying PDE has
a conservation form, it is also possible for the collocation scheme to satisfy
an analogous discrete conservation law. This procedure is implemented in
MOVCOL and is described in detail in Huang and Russell (1996) The resulting
ODEs are somewhat stiff, and are typically solved using an appropriate
stiff solver such as an SDIRK (singly diagonally implicit Runge–Kutta) or
a BDF (backward differentiation formula) method. In MOVCOL they are
solved using a BDF method in the code dassl (Petzold 1982).
3.1.4. Spectral methods
Spectral methods provide an attractive alternative to finite difference and
finite element methods for numerical solution of PDEs. They involve approximation
by global basis functions, such as trigonometric or algebraic
polynomials. For problems with smooth solutions the convergence rate of
spectral methods is faster than algebraic, as the number of grid points increases,
and the significance of this so-called spectral convergence is that
a specified accuracy can usually be achieved using fewer grid points than
would be required by the algebraically convergent finite difference or finite
element approaches. However, if a solution has a steep region such as a
boundary layer or an interior layer, spectral methods will achieve high accuracy
only if the number of grid points is sufficiently high to permit resolution
of the localized phenomena. To overcome this difficulty, a common approach
is to apply a coordinate transformation that is designed to smooth out regions
of high gradient. Such a transformation can be generated numerically
and adaptively through a moving mesh method. Another benefit of using
a moving mesh method is that PDEs can be conveniently discretized on
the computational domain where a rectangular or cubic mesh is often used.
Adaptivity of this type has proved successful, producing highly accurate
solutions to problems that have steep, smooth solutions using a reasonably
small number of grid points, although some care must be taken that the numerically
generated coordinate transformation should be made sufficiently