chapter  5
32 Pages

Variable Order DQ Method

Ever since their introduction, DQ methods have attracted attention from researchers worldwide and have triggered a variety of applications to engineering problems (Bellman et al. 1972; Shu, 2000a; Bert et al., 1996a,b,c). Most of the applications are relevant to static or free vibrations. Recent years saw increasing applications of DQ methods to dynamic problems (Fung, 2001; Shu and Kha, 2002). In the dynamic analysis, however, dynamic numerical instability might become a serious problem. As time marches, accumulations of numerical errors may deteriorate the accuracy of the solutions. Numerical stability is an important factor when dynamic problems are studied by using DQ method. First in this chapter, a simple numerical example is given to demonstrate

dynamic numerical instability associated with DQ discretization. It is revealed from the example that grid points near and on boundaries exert dominant influence on dynamic numerical instability, as confirmed by various researchers (Moradi and Taheri, 1998; Quan and Chang, 1989a, b; Bert and Malik, 1996a; Shu et al., 2001). This finding led to the proposal of variable order DQ method in which we distinguish two main classes of nodes (grid points), core nodes and cortical nodes according to their distance from boundaries (Zong, 2003b). Variable order DQ approximations are applied to core and cortical nodes. At core nodes, higher order DQ schemes are employed while at cortical nodes lower order DQ schemes are applied. This variable order approach turns out to be very effective in keeping the balance between the dynamic stability and accuracy. Numerical examples manifest that the approach introduced in this chapter is applicable to linear and highly nonlinear dynamic equations. Numerical instability associated with dynamic analysis may be improved

through use of variable DQ method for spatial discretization or use of a precision integration technique for temporal discretization. This is the content

Consider a continuous function f(x, t) defined in terms of time t and space x. Suppose on the following N discrete spatial grid points (or nodes)

x1 < x2 < . . . < xN (5.1)

the values of the function f(x, t) are

f(x1, t), f(x2, t), . . . , f(xN , t) (5.2)

The DQ discretization of the r-th order partial derivative with respect to x is given by the following equation as repeatedly quoted

∂rf(xi, t)

∂xr =

e (r) ij f(xj , t), i = 1, 2, . . .N, r = 1, 2, . . .N − 1 (5.3)

where e (r) ij is the weighting coefficient for the r-th order derivative with respect

to x as defined by Eq.(1.46). The derivatives are dependent on the function values on grid points and on spatial grid spacing. For a dynamic equation, we also need to discretize time. There have been

attempts to apply DQ approach for temporal discretization (see Chapter 1), but in this chapter we employ simpler temporal discretization method, fourthorder Runge−Kutter (RK) scheme (see Chapter 8). To show dynamic numerical instability we first consider the following string

vibration equation

∂2u(x, t)

∂t2 = ∂2u(x, t)

∂x2 , 0 ≤ x ≤ 1 (5.4)

subject to the following initial and boundary conditions

u(x, t = 0) = sin(πx), u(x = 0, t) = u(x = 1, t) = 0 (5.5)

The analytical solution to the above equations is known to be

u(x, t) = sin(πx) cos(πt) (5.6)

To apply RK method, we rewrite Eq. (5.4) into the following form of a set of first-order ordinary differential equations

 


, i = 1, . . . , N (5.7)

FIGURE 5.1: Dynamic instability in RK-DQ discretization of string vibration equation.