Error Estimation for the Time to a Threshold Value for the Shallow Water Equations

  • Chaudhry, Jehanzeb (University of New Mexico)
  • Estep, Donald (Simon Fraser University)
  • Giannini, Trevor (University of New Mexico)
  • Stevens, Zachary (University of New Mexico)
  • Tavener, Simon (Colorado State University)

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Accurate estimation of the error in a quantity-of-interest (QoI) is a fundamental requirement for the robust application of numerical simulations in science and engineering. Adjoint based error estimates for QoIs modeled as bounded linear functionals (or linearized as such) have long been developed. However, the time at which a functional of the solution achieves a threshold value, which we refer to as the time to an ``event'', is often the primary concern in a physical system. An important example of this is the time at which a tsunami wave crosses a certain threshold height. In this work we consider such problems. Unfortunately, the error in an estimated time to a particular event cannot be quantified using standard approaches. Taylor’s theorem and an adjoint-based a posteriori analysis are utilized to derive an accurate error estimate for this non-standard non-linear QoI. While the analysis applies to a large class of evolutionary differential equations, this talk focuses on its application to the Shallow Water Equations which models a tsunami wave. The accuracy of the estimate is demonstrated on a number of setups of varying initial conditions and bottom ocean topographies.