Sensitivity Analysis for Shallow Water Equations - Advanced Study ...

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SE-1D: Space Discretization Regular Element I j Reference Element ! x j!1/2 x j+1/2 ! 1 GLL Grid +1 How do the neighboring elements talk? Nodal - Averaging the values on the element boundaries Time Integration Scheme Leapfrog Scheme u n+1 = u n−1 − 2∆tF (u n ) The solution within each element is interpolated with a high-order polynomial using nodal basis set. Vani Cheruvu (ASP Research Review) Adjoint Sensitivity Analysis 06/14/2007 18 / 30
SFV in 1D For a given order of accuracy k, each SV is divided into k control volumes (CV) The ith CV of jth SV is : C j,i = [x j,i−1/2 , x j,i+1/2 ] and with cell width ∆x j,i = x j,i+1/2 − x j,i−1/2 x CV j,i!1/2 x j,i+1/2 x j!1/2 SV I j x j+1/2 Discontinuous Galerkin: Flux evaluation - Riemann Solver Spectral Finite Volume: Numerical flux at element boundaries and analytical flux at the cell boundaries Vani Cheruvu (ASP Research Review) Adjoint Sensitivity Analysis 06/14/2007 19 / 30
Page 1 and 2: Sensitivity Analysis for Shallow Wa
Page 3 and 4: Vani Cheruvu (ASP Research Review)
Page 5 and 6: Chennai’s coastline before and af
Page 7 and 8: Chennai’s Marina beach after tsun
Page 9 and 10: Why an Adjoint Equation? Diffusion
Page 11 and 12: Shallow Water Equations Objective F
Page 13 and 14: Adjoint Sensitivity Method Derivati
Page 15 and 16: Solution of Shallow Water Equations
Page 17: Spectral Element Method The domain
Page 21 and 22: 2D Grid X X X X X X CV X X X X X X
Page 23 and 24: Nonreflective Outflow Boundary Adjo
Page 25 and 26: Neighboring elements that share sam
Page 27 and 28: Gibbs Phenomenon (sin(x) + 1 3 sin(
Page 29 and 30: DG-1D : Numerical Examples U t + F

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