The motivation for the comparison of these spectral methods is to compute solutions to high-order semi linear initial boundary value problems found in non linearly elastic models for micro structure formation during phase transitions in which a small Ginzburg or capillarity term is added. Typical examples of these semi linear initial boundary value problems can be found in studies by Ahluwalia, Lookman and Saxena [2], Hoffman and Rybka [26, 27] and Truskinovsky [48, 49]. A finite difference numerical study in a scalar model of microstructure formation in two dimensions without a capillarity term by Swart and Holmes [45] shows the development of wrinkles at one fixed boundary, these wrinkles then coarsen into the interior of the computational domain. It is thought that the errors in the finite difference discretization approximation used in this simulation introduced an effective capillarity term which prompted Kohn and Müller [31, 32] and Conti [9] to provide analytical explanations for the multiscale behavior by considering the form of the global minimizer of the associated energy with a capillarity term. Basinski and Christian [4] have also predicted that multiscale wrinkling behavior should be observed in experiments. However, high spatial resolution Fourier spectral collocation studies of microstructure formation in elasticity by Ahluwalia and Ananthakrishna [1], Kerret al. [29], Killough [30] and Lookman et al. [33] which use periodic boundary conditions do not obtain branched multiscale micro structures in the final steady-state solution. To performing large scale simulations of semi-linear initial boundary value problems, an easily parallelizable time stepping routine should be combined with an easily paralleizable spatial discretization with non-periodic boundary conditions.
This precludes the use of high temporal accuracy spectral and exponential integration schemes which are still under development, and for present day simulations, implicit-explicit time marching scheme should be used together with a high resolution spatial discretization. In an implicit-explicit time marching scheme, not only is the function required at each time step, but derivatives of the function may also be required to calculate the explicit part of the time step. Thus it is worth comparing the accuracy and computational cost of solutions and derivatives of solutions obtained by using the spectral pre-conditioned Chebyshev tau and spectral integration Chebyshev collocation methods in solving linear boundary value problems because in a typical IMEX scheme, a linear boundary value problem is solved at each time step. The study shows that the Chebyshev integration method allows one to efficiently simulate two scale microstructures and should be considered as a viable alternative to low spatial accuracy finite difference (for example used by Ahluwalia, Lookman and Saxena [2], Ahluwalia et al. [3] and by Vainchtein [50]) and finite element methods (for example used by Dondl and Zimmer [16] and reviewed in Luskin [35]) to model microstructure formation in continuum models with higher order derivatives with non-periodic boundary conditions.
Only a few studies of microstructure formation have used spectral spatial discretizations that allow for non-periodic boundary conditions. Yao and Gooding [53] found a method of obtaining high resolution solutions to one-dimensional, fourth-order in space initial boundary value models for phasetransitions by splitting the spatial domain into seperate portions and using Hermite polynomials in each interval. They were unable to use a largenumber of modes in their study. Reid and Gooding [41, 42] used a Chebyshev spectral implementation to be able to enforce a wide variety of boundary conditions, but they could only perform low resolution simulations with their Galerkin type method. Lyons et al. [36] have also obtained a fast spectral algorithm for non-periodic boundary conditions, but have implemented it only for second-order in space boundary value problems.
Chebyshev collocation methods can also be used to obtain high resolution numerical solutions to the KdV, Allen-Cahn and Cahn-Hilliard equations (see for example Xu and Tang [52] and Kassam and Trefethen [28]). They may also be useful in examining viscosity solutions to conservation laws, such as Burgers equation regularized by viscosity and dispersion, see, for example, Schonbeck [43] , Chen, Du and Tadmor [7] or Hesthaven, Gottlieb and Gottlieb [25]. Here numerical simulations can indicate the existence of possible vanishing viscosity or vanishing dispersion limits in bounded domains. Finally, as explained by Mai-Duy [37] and Mai-Duy and Tanner [38], this method can also be used to solve the biharmonic equation which often occurs in thin plate models and in the vorticity-streamfunction formulation of the Navier-Stokes equations.
