# Dual weighted residual method e weighted residual of ... tions in the dual problem. Discretization...

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Adaptive mesh refinement + quantities of interest computed by goal functional ⇒ Dual weighted residual method

ηe = weighted residual of primal problem + weighted residual of dual problem

Primal problem: 2D nonlinear Shallow water equations (SWE) Dual problem: adjoint equations of SWE depending on solutions of primal problem

Discretization: Runge-Kutta discontinuous Galerkin scheme ⇒ spatial discontinuous primal solutions ⇒ discontinuous coefficients in dual problem Riemann problem with discontinuous coefficients

1 Averaged coefficients + usual Riemann solver 2 Weighted method (upwind/downwind)

Riemann solver for the adjoint shallow water equations with discontinuous coefficients

Susanne Beckers1,3, Jörn Behrens2,3, Winnifried Wollner2,3 1 Lothar Collatz School for Computing in Science, 2 Lothar Collatz Center for Computing in Science, 3 Department of Mathematics

All three belong to the University of Hamburg, Germany

Project Aims and Challenges

A strategy for increasing accuracy and efficiency in numerical simulations is adaptive mesh refinement. In order to control the area of refinement, in particular for special quantities of interest, a goal functional is defined. An optimization method of choice is the dual weighted residual method (DWR), in which the element wise error estimator ηe is given by weighted residuals of the primal and the dual computation. The dual run solves the adjoint equations with solutions of the forward (primal) computation as coefficients. We solve the 2D non-linear shallow water equations (SWE) by a Runge-Kutta discontinuous Galerkin (DG) scheme, which leads to spatial discontinuous solutions. Thus, the adjoint fluxes feature discontinuous coefficients, which pose numerical problems to the corresponding Riemann solvers. One approach is to average the coefficients of the adjacent elements and solve a usual Rie- mann problem for example with the Rousanov numerical flux. An other approach is a weighted method that works as an upwind or downwind scheme, depending on the coefficients.

Primal Model

The SWE for velocities u = (u(x, y, t), v(x, y, t))T and displacement H = h̄(x, y) +h(x, y, t) are given as

∂tu + [u · ∇] u + k× fu + g∇h = 0, ∂th +∇ · [uH ] = 0,

with f, g, h̄(x, y) ∈ R the Coriolis force, the gravity and the average height, respectively. Furthermore, we set initial conditions as

u(x, y, 0) = u0(x, y) ∀(x, y) ∈ Ω, v(x, y, 0) = v0(x, y) ∀(x, y) ∈ Ω, h(x, y, 0) = h0(x, y) ∀(x, y) ∈ Ω,

and periodic boundary conditions on the boundary of Ω = [0, 100] × [0, 100]. The initial velocity is given as a quasi stationary vortex with an underlying drift in x direction.

Left: Initial height in [103m] Right: Initial magnitude of velocity in [m/s].

Dual Model

A goal functional which is evaluated at final time t = T leads, via a minimization problem where u, v and h are subject to the primal problem, to the dual problem:

−∂tz− [ u× · ∇

] z× −∇z1u−∇z2v − k× fz× −∇ [z3H ] = 0,

−∂tz3 − g∇ · z− 2u · ∇z3 = 0,

where u× = (v(x, y, t), u(x, y, t))T is the vector of velocities with switched components, z = (z1(x, y, t), z2(x, y, t))

T the sensitivities of the velocities, z× = (z2(x, y, t), z1(x, y, t))T its switched vector and z3(x, y, t) the sensitivity of the vertical displacement. Taking the potential energy as goal functional evaluated at time t = T ,

J(x, y, T ) = 1

|Ω|

∫ Ω

gh (x, y, T ) dx,

leads to dual initial conditions as

z1(x, y, T ) = 0 ∀(x, y) ∈ Ω, z2(x, y, T ) = 0 ∀(x, y) ∈ Ω, z3(x, y, T ) =

gh(x, y, T )

|Ω| ∀(x, y) ∈ Ω.

The spatial periodic boundary conditions in the primal problem result also in periodic condi- tions in the dual problem.

Discretization and mesh

•Time discretization: Runge-Kutta method for K time steps • Spatial discretization: DG method on elements e ∈ Th

– Rousanov Riemann solver for the primal problem

– Averaged Rousanov or weighted scheme Riemann solver for the dual problem

•Global uniform mesh for primal and dual run •Refinement by bisection according to the error estimator ηe =

∑K k=0 ηe,k

For U = (u, v, h) and Z = (z1, z2, z3) holds

|J(U)− J(Uh)| ≤ c (

1

2 ρ(Uh; Zh/2 − Zh) +

1

2 ρ∗(Zh; Uh/2 −Uh)

) = c

∑ e∈Th

ηe

Riemann Solvers for the Dual Problem

The dual problem can be rearranged as

−∂tVdual +

−2u−2v −2u

∂x(Vdual) + −2u−2v −2v

∂y(Vdual) = Sdual(Vdual) where Vdual = (z1z3, z2z3, z3) is the conserved quantity and ∇ ·F = ∂xVdual + ∂yVdual is the divergence of the flux F. Averaged Rousanov solver

1. Projection to normal direction for element e = left (l), right (r) and i = 1, 2, 3: Speed of sound Coefficients

λe,i = −2(uen1 + ven2) ce,1 = −2uen1 ce,2 = −2ven2 ce,3 = −2(uen1 + ven2)

2. Averaging

λ = λl + λr

2 , cav =

cl + cr 2

3. Flux on the edge E between left and right element

FE(Vdual,l,r) = (F(Vdual,l) + F(Vdual,r)) · cav + λ(Vdual,l −Vdual,r)

2

4. Projection of the flux to normal direction

Fnormal = FE(Vdual,l,r) · n

Left: Dual height after 50 time steps with the averaged Rousanov solver Right: Dual magnitude of velocity after 50 time steps with the averaged Rousanov solver

Weighted scheme for discontinuous coefficients

1. Projection to normal direction Speed of sound Dual quantities

λe,i = −2(uen1 + ven2) de,1,2 = ze,1n1 + ze,2n2 de,3 = z3

2. Flux on the edge E between left and right element:

Fi,E(Vdual,l,r) = λildir + λirdil λil + λir

, i = 1, 2, 3

3. Projection of the flux to normal direction

Fnormal = FE(Vdual,l,r) · n

Left: Dual height after 50 time steps obtained with the weighted scheme Right: Dual magnitude of velocities after 50 time steps obtained with the weighted scheme The velocities are not correctly transported in y-direction but the features resemble the aver- aged Rousanov scheme. The dual height is a result of the incorrect transport.

Selected References

[1] R. Becker, R. Rannacher: An optimal control approach to a posteriori error estimation in finite element methods, Acta Numerica 2001, pp. 1-102 (2001)

[2] F. X. Giraldo, M. Restelli: High-order semi-implicit time-integrators for a triangular discontinuous Galerkin oceanic shallow water model, Int. J. Numer. Meth. Fluids 63 (9), pp. 1077-1102 (2010)

[3] L. Remaki: Riemann solution for hyperbolic equations with discontinuous coeffi- cients, Conference proceedings: Applications of Mathematics 2013, ISBN 978-80-85823- 61-5, pp. 188-196 (2013)

Acknowledgement

This work is partly supported by Forschungs- und Wissenschaftsstiftung Hamburg.

Riemann solver for the adjoint shallow water equations with discontinuous coefficients

Susanne Beckers1,3, Jörn Behrens2,3, Winnifried Wollner2,3 1 Lothar Collatz School for Computing in Science, 2 Lothar Collatz Center for Computing in Science, 3 Department of Mathematics

All three belong to the University of Hamburg, Germany

Project Aims and Challenges

A strategy for increasing accuracy and efficiency in numerical simulations is adaptive mesh refinement. In order to control the area of refinement, in particular for special quantities of interest, a goal functional is defined. An optimization method of choice is the dual weighted residual method (DWR), in which the element wise error estimator ηe is given by weighted residuals of the primal and the dual computation. The dual run solves the adjoint equations with solutions of the forward (primal) computation as coefficients. We solve the 2D non-linear shallow water equations (SWE) by a Runge-Kutta discontinuous Galerkin (DG) scheme, which leads to spatial discontinuous solutions. Thus, the adjoint fluxes feature discontinuous coefficients, which pose numerical problems to the corresponding Riemann solvers. One approach is to average the coefficients of the adjacent elements and solve a usual Rie- mann problem for example with the Rousanov numerical flux. An other approach is a weighted method that works as an upwind or downwind scheme, depending on the coefficients.

Primal Model

The SWE for velocities u = (u(x, y, t), v(x, y, t))T and displacement H = h̄(x, y) +h(x, y, t) are given as

∂tu + [u · ∇] u + k× fu + g∇h = 0, ∂th +∇ · [uH ] = 0,

with f, g, h̄(x, y) ∈ R the Coriolis force, the gravity and the average height, respectively. Furthermore, we set initial conditions as

u(x, y, 0) = u0(x, y) ∀(x, y) ∈ Ω, v(x, y, 0) = v0(x, y) ∀(x, y) ∈ Ω, h(x, y, 0) = h0(x, y) ∀(x, y) ∈ Ω,

and periodic boundary conditions on the boundary of Ω = [0, 100] × [0, 100]. The initial velocity is given as a quasi stationary vortex with an underlying drift in x direction.

Left: Initial height in [103m] Right: Initial magnitude of velocity in [m/s].

Dual Model

A goal functional which is evaluated at final time t = T leads, via a minimization problem where u, v and h are subject to the primal problem, to the dual problem:

−∂tz− [ u× · ∇

] z× −∇z1u−∇z2v − k× fz× −∇ [z3H ] = 0,

−∂tz3 − g∇ · z− 2u · ∇z3 = 0,

where u× = (v(x, y, t), u(x, y, t))T is the vector of velocities with switched components, z = (z1(x, y, t), z2(x, y, t))

T

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