A well-balanced reconstruction of wet/dry fronts for the shallow water equations Guoxian Chen Wuhan University, P.R.China Co-authors: Bollermann, Kurganov, Noelle May 24, 2014
Outline Governing Equations 1 A Central-Upwind Scheme for the Shallow Water Equations 2 A New Reconstruction at the Almost Dry Cells 3 Positivity Preserving and Well-Balancing 4 Numerical Experiments 5 Conclusion and Future Worker 6 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 2 / 38
Governing Equations 1 A Central-Upwind Scheme for the Shallow Water Equations 2 A New Reconstruction at the Almost Dry Cells 3 Positivity Preserving and Well-Balancing 4 Numerical Experiments 5 Conclusion and Future Worker 6 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 3 / 38
Saint-Venant system of shallow water equations Notations: h ( x , t ) is the fluid depth, u ( x , t ) is the velocity, g is the gravitational constant, B ( x ) represents the bottom topography. Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38
Saint-Venant system of shallow water equations Notations: h ( x , t ) is the fluid depth, u ( x , t ) is the velocity, g is the gravitational constant, B ( x ) represents the bottom topography. In one dimension, the Saint-Venant system reads: h t + ( hu ) x = 0 , � 2 gh 2 � hu 2 + 1 ( hu ) t + x = − ghB x , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38
Saint-Venant system of shallow water equations Notations: h ( x , t ) is the fluid depth, u ( x , t ) is the velocity, g is the gravitational constant, B ( x ) represents the bottom topography. In one dimension, the Saint-Venant system reads: h t + ( hu ) x = 0 , � 2 gh 2 � hu 2 + 1 ( hu ) t + x = − ghB x , subject to the initial conditions h ( x , 0) = h 0 ( x ) , u ( x , 0) = u 0 ( x ) , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 4 / 38
Numerical difficulties Figure: Numerical storm over lake Rursee, produced by a naive finite volume scheme Quasi steady solutions, Coarse grid, Numerical storm etc. Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 5 / 38
Numerical difficulties w B j ˜ B B x j − 1 x j + 1 2 2 Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0 , w := h + B = Const . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38
Numerical difficulties w B j ˜ B B x j − 1 x j + 1 2 2 Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0 , w := h + B = Const . dry areas (island, shore). Lose strictly hyperbolic in the dry areas ( h = 0): u ± √ gh ; The calculation will simply due to h < 0 break down. Solution: a positive preserving scheme h ≥ 0 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38
Numerical difficulties w B j ˜ B B x j − 1 x j + 1 2 2 Quasi steady solutions, Coarse grid, Numerical storm etc. Solution: Well-balanced scheme u = 0 , w := h + B = Const . dry areas (island, shore). Lose strictly hyperbolic in the dry areas ( h = 0): u ± √ gh ; The calculation will simply due to h < 0 break down. Solution: a positive preserving scheme h ≥ 0 “dry lake at rest” steady state: hu = 0 , h = 0. Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 6 / 38
Governing Equations 1 A Central-Upwind Scheme for the Shallow Water Equations 2 A New Reconstruction at the Almost Dry Cells 3 Positivity Preserving and Well-Balancing 4 Numerical Experiments 5 Conclusion and Future Worker 6 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 7 / 38
Overview of KP scheme: Kurganov and Petrova, 2007 w B j ˜ B B x j − 1 x j + 1 2 2 Bottom is Continuous, piecewise linear approximated by � B � � x − x j − 1 � B ( x ) = B j − 1 2 + 2 − B j − 1 · 2 , 2 ≤ x ≤ x j + 1 2 . B j + 1 x j − 1 ∆ x 2 where B ( x j + 1 2 + 0) + B ( x j + 1 2 − 0) 2 := , B j + 1 2 then � B j + 1 2 + B j − 1 1 B j := � � 2 B ( x j ) = B ( x ) dx = . ∆ x 2 I j Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 8 / 38
Central-upwind semi-discretization Cell averages of U := ( w , hu ) T is defined � 1 U j ( t ) ≈ U ( x , t ) dx . ∆ x I j Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38
Central-upwind semi-discretization Cell averages of U := ( w , hu ) T is defined � 1 U j ( t ) ≈ U ( x , t ) dx . ∆ x I j Time-dependent ODEs (SSP-RK-3) H j + 1 2 ( t ) − H j − 1 2 ( t ) d dt U j ( t ) = − + S j ( t ) , ∆ x Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38
Central-upwind semi-discretization Cell averages of U := ( w , hu ) T is defined � 1 U j ( t ) ≈ U ( x , t ) dx . ∆ x I j Time-dependent ODEs (SSP-RK-3) H j + 1 2 ( t ) − H j − 1 2 ( t ) d dt U j ( t ) = − + S j ( t ) , ∆ x the cell averages of the source term: � 1 S := (0 , − ghB x ) T . S j ( t ) ≈ S ( U ( x , t ) , B ( x )) dx , ∆ x I j i.e 2 − B j − 1 B j + 1 (2) S ( t ) := − gh j 2 . j ∆ x Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 9 / 38
central-upwind numerical fluxes The central-upwind numerical fluxes: a + 2 F ( U − 2 ) − a − 2 F ( U + 2 , B j + 1 2 , B j + 1 2 ) j + 1 j + 1 j + 1 j + 1 2 ( t ) = H j + 1 a + 2 − a − j + 1 j + 1 2 a + 2 a − � � j + 1 j + 1 U + 2 − U − + 2 , a + 2 − a − j + 1 j + 1 2 j + 1 j + 1 2 where we use the following flux notation: � � T hu , ( hu ) 2 w − B + g 2 ( w − B ) 2 F ( U , B ) := . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 10 / 38
central-upwind numerical fluxes The central-upwind numerical fluxes: a + 2 F ( U − 2 ) − a − 2 F ( U + 2 , B j + 1 2 , B j + 1 2 ) j + 1 j + 1 j + 1 j + 1 2 ( t ) = H j + 1 a + 2 − a − j + 1 j + 1 2 a + 2 a − � � j + 1 j + 1 U + 2 − U − + 2 , a + 2 − a − j + 1 j + 1 2 j + 1 j + 1 2 where we use the following flux notation: � � T hu , ( hu ) 2 w − B + g 2 ( w − B ) 2 F ( U , B ) := . the local speeds a ± 2 in numerical flux are obtained using the eigenvalues of j + 1 the Jacobian ∂ F ∂ U as follows: � � � � a + u + gh + 2 , u − gh − 2 = max 2 + 2 + 2 , 0 , j + 1 j + 1 j + 1 j + 1 j + 1 � � � � a − u + gh + 2 , u − gh − 2 = min 2 − 2 − 2 , 0 . j + 1 j + 1 j + 1 j + 1 j + 1 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 10 / 38
Initial date reconstruction(Surface Gradient Method) piecewise linear reconstruction q stands for w and u respectively. � q ( x ) := q j + ( q x ) j ( x − x j ) , x j − 1 2 < x < x j + 1 2 , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38
Initial date reconstruction(Surface Gradient Method) piecewise linear reconstruction q stands for w and u respectively. � q ( x ) := q j + ( q x ) j ( x − x j ) , x j − 1 2 < x < x j + 1 2 , The generalized minmod limiter: � � θ q j − q j − 1 , q j +1 − q j − 1 , θ q j +1 − q j ( q x ) j = minmod , θ ∈ [1 , 2] , ∆ x 2∆ x ∆ x with min j { z j } , if z j > 0 ∀ j , minmod ( z 1 , z 2 , ... ) := max j { z j } , if z j < 0 ∀ j , 0 , otherwise , Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38
Initial date reconstruction(Surface Gradient Method) piecewise linear reconstruction q stands for w and u respectively. � q ( x ) := q j + ( q x ) j ( x − x j ) , x j − 1 2 < x < x j + 1 2 , The generalized minmod limiter: � � θ q j − q j − 1 , q j +1 − q j − 1 , θ q j +1 − q j ( q x ) j = minmod , θ ∈ [1 , 2] , ∆ x 2∆ x ∆ x with min j { z j } , if z j > 0 ∀ j , minmod ( z 1 , z 2 , ... ) := max j { z j } , if z j < 0 ∀ j , 0 , otherwise , average velocity is defined as with ǫ = 10 − 9 � ( hu ) j / h j , if h j ≥ ǫ, u j := 0 , otherwise . Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38
Initial date reconstruction(Surface Gradient Method) piecewise linear reconstruction q stands for w and u respectively. � q ( x ) := q j + ( q x ) j ( x − x j ) , x j − 1 2 < x < x j + 1 2 , The generalized minmod limiter: � � θ q j − q j − 1 , q j +1 − q j − 1 , θ q j +1 − q j ( q x ) j = minmod , θ ∈ [1 , 2] , ∆ x 2∆ x ∆ x with min j { z j } , if z j > 0 ∀ j , minmod ( z 1 , z 2 , ... ) := max j { z j } , if z j < 0 ∀ j , 0 , otherwise , average velocity is defined as with ǫ = 10 − 9 � ( hu ) j / h j , if h j ≥ ǫ, u j := 0 , otherwise . water height is recontructed by h ± 2 = w ± 2 − B j + 1 2 . j + 1 j + 1 Guoxian Chen (WHU) well-balanced reconstruction of wet/dry fronts May 24, 2014 11 / 38
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