Solving initial-boundary value problems without numerical differentiation

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The numerical solution of nonlinear partial differential equations with nontrivial boundary conditions is central to many areas of modelling. When high accuracy is required (pseudo) spectral methods are usually the first choice. Typically in this approach we search for the pre-image under a linear operator which represents a combination of spatial derivatives along with the boundayr conditions in every time step. This operator can be quite ill-conditioned. On a basis of Chebyshev polynomials for instance the condition number increases algebraically with the number of basis functions. I will present an alternative method based on recent work by Viswanath and Tobasco which avoids numerical differentiation entirely through the use of Green's functions. I will demonstrate this method on the Kuramoto-Sivashinsky equation with fixed boundary conditions.