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Reduced Order Models for Parameterized Hyperbolic Conservation Laws with Shock Reconstruction

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Continued advances in high performance computing are enabling researchers in computational science to simulate more complex physical models. Such simulations can occupy massive supercomputers for extended periods of time. Unfortunately, the cost of these complex simulations renders parameter studies (e.g., design optimization or uncertainty quanti cation) infeasible, where multiple simulations must be run to explore the space of design parameters or uncertain inputs. A common fix is to construct a cheaper reduced order model—trained on the outputs of a few carefully selected simulation runs—for use in the parameter study.

Model reduction for large-scale simulations is an active research field. Techniques such as reduced basis methods and various interpolation schemes have been used successfully to approximate the simulation output at new parameter values at a fraction of the computational cost of a full simulation. These methods perform best when the solution is smooth with respect to the model parameters. The solution of nonlinear conservation laws are known to develop discontinuities in space even for smooth initial data. These spatial discontinuities typically imply discontinuities in the parameter space, which severely diminish the performance of standard model reduction methods.

We present a method for constructing an accurate reduced order model of the solution to a parameterized, nonlinear conservation law. We use a standard method for an initial guess and propose a metric for determining regions in space/time where the standard method yields a poor approximation. We then return to the conservation law and correct the regions of low accuracy. We will describe the method in general and present results on the inviscid Euler equations with parameterized initial conditions.

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