SYMPOSIUM MINISYMPOSIA

MINISYMPOSIUM ABSTRACT

We use a modal footprint in frequency and amplitude space to reduce total simulation error in numerical simulations based on finite difference approximations. We then approximate non-linear terms based on the amplitude of these modes by varying the derivative terms, rather than the undifferentiated terms, to derive a very accurate, fully non-linear approximation that is not only explicit, but efficient as well. Development time for each implementation is significantly longer, but the resulting numerical approximation follows the behavior of the underlying partial differential equations more accurately.

To illustrate our ideas, imagine two small amplitude waves interacting. Their interaction will be essentially linear. But for two large amplitude waves, strongly non-linear interactions may occur as the waves pass through each other. Hence it is natural to model non-linear interactions in amplitude and frequency space. But it is expensive. Using a method of undetermined coefficients for both construction of the finite difference scheme to gain group velocity accuracy and for post-processing to gain amplitude accuracy allows us to construct a finite difference method that has very low error in an a priori specified bandwidth. We assume that this bandwidth is where the strong non-linear interactions will occur for the physics modeled. We then construct a set of fitting functions throughout amplitude and frequency space that optimizes our numerical approximations for the differentiated non-linear term. The structure of the fitting functions, and the astonishing performance improvements achieved, have convinced the authors that this is a fruitful area that needs to be further examined.

The non-linear change of emphasis from undifferentiated to differentiated terms is justified via a convergence proof. Along with this proof, we will introduce a mathematical framework to consider non-linear dynamics in frequency and amplitude space.

We will present examples with the viscous Burgers equation, comparing our method to other methods know to perform well on this problem. Our results are significant improvements over the methods with which we compare.

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To obtain more information about the meeting send e-mail to: fscc98@pax.st.usm.edu.

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