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Recent progress has reduced the study of nonlinear time-evolutionary stability of viscous shock waves to determination of spectral stability, or location of eigenvalues of the linearized operator about the wave: that is, from a rather stiff PDE initial value problem to an ODE boundary-value problem that is analytically and numerically well conditioned. However, the resulting ODE are variable-coefficient and dimension three or higher, so challenging in their own right. Their study presents a new array of problems rather different from those of the preceding theory. We discuss general strategies for their solution combining numerical methods with methods from asymptotic ode/singular perturbation theory, and give specific examples indicating their use.