Well-balanced asymptotic preserving schemes for singular limit flows - PowerPoint PPT Presentation

Well-balanced asymptotic preserving schemes for singular limit flows Mária Lukáčová Johannes Gutenberg-Universität Mainz K.R. Arun (Trivandrum), S. Noelle (RWTH Aachen), L. Yelash & G. Bispen (JGU Mainz), A. Müller & F. Giraldo (Monterey)

Application Meteorology: Cloud Simulation Gravity induces hydrostatic balance How do clouds evolve over long periods of time? M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 1 / 49

Multiscale phenomena of oceanographical, atmospherical flows -wave speeds differ by several orders: � u � << c ⇒ M , Fr : = � u � << 1 c -typically Fr ≈ 10 − 2 M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 2 / 49

Multiscale phenomena of oceanographical, atmospherical flows -wave speeds differ by several orders: � u � << c ⇒ M , Fr : = � u � << 1 c -typically Fr ≈ 10 − 2 max ( | u | + c , | v | + c ) ∆ t ≤ 1 ∆ x M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 2 / 49

Multiscale phenomena of oceanographical, atmospherical flows -wave speeds differ by several orders: � u � << c ⇒ M , Fr : = � u � << 1 c -typically Fr ≈ 10 − 2 max ( | u | + c , | v | + c ) ∆ t ≤ 1 ∆ x � ∆ t �� � � 1 + 1 u 2 + v 2 ∆ x ≤ 1 max Fr - number of time steps O ( 1/ Fr ) -low Mach / low Froude number problem [ Bijl & Wesseling (’98), Klein et al.(’95, ’01), Meister (’99,01), Munz &Park (’05), Degond et al. (’11) . . . ] M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 2 / 49

Accuracy problem - numerical viscosity of upwind methods depends on Fr - truncation error grows as Fr → 0 [Guillard, Viozat (’99), Rieper (’10)] - AIM: reduce adverse effect of 1 + 1/ Fr large time step scheme: ∆ t does not depends on Fr efficient scheme for advection effects stability and accuracy of the scheme is independent on Fr M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 5 / 49

Asymptotic preserving schemes Goal: Derive a scheme, which gives a consistent approximation of the limiting equations for ε = Fr → 0 [ S.Jin&Pareschi(’01), Gosse&Toscani(’02), Degond et al.(’11), . . . ] M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 6 / 49

Asymptotic preserving schemes Goal: Derive a scheme, which gives a consistent approximation of the limiting equations for ε = Fr → 0 [ S.Jin&Pareschi(’01), Gosse&Toscani(’02), Degond et al.(’11), . . . ] - to illustrate the idea: shallow water eqs. - z = h + b , h - water depth, z - mean sea level to the top surface, b - mean sea level to the bottom ( b ≤ 0 ) M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 6 / 49

Asymptotic preserving schemes Goal: Derive a scheme, which gives a consistent approximation of the limiting equations for ε = Fr → 0 [ S.Jin&Pareschi(’01), Gosse&Toscani(’02), Degond et al.(’11), . . . ] - to illustrate the idea: shallow water eqs. - z = h + b , h - water depth, z - mean sea level to the top surface, b - mean sea level to the bottom ( b ≤ 0 ) ∂ t z + ∂ x m + ∂ y n = 0 1 1 ∂ t m + ∂ x ( m 2 / ( z − b )) + ∂ y ( mn / ( z − b )) + 2 Fr 2 ∂ x ( z 2 ) = Fr 2 b ∂ x z 1 1 ∂ t n + ∂ x ( mn / ( z − b )) + ∂ y ( n 2 / ( z − b )) + 2 Fr 2 ∂ y ( z 2 ) = Fr 2 b ∂ y z M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 6 / 49

Asymptotic expansion -rigorous analysis [Klainerman & Majda (’81), Feireisl & Novotn´ y (2009, 2013)] -formally: ( ε = Fr ) z ε ( x , t ; ε ) = z ( 0 ) ( x , t ) + ε z ( 1 ) ( x , t ) + ε 2 z ( 2 ) ( x , t ) u ε ( x , t ; ε ) = u ( 0 ) ( x , t ) + ε u ( 1 ) ( x , t ) + ε 2 u ( 2 ) ( x , t ) M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 7 / 49

Asymptotic expansion -rigorous analysis [Klainerman & Majda (’81), Feireisl & Novotn´ y (2009, 2013)] -formally: ( ε = Fr ) z ε ( x , t ; ε ) = z ( 0 ) ( x , t ) + ε z ( 1 ) ( x , t ) + ε 2 z ( 2 ) ( x , t ) u ε ( x , t ; ε ) = u ( 0 ) ( x , t ) + ε u ( 1 ) ( x , t ) + ε 2 u ( 2 ) ( x , t ) plug into the SWE = ⇒ z ( 0 ) = z ( 0 ) ( t ) ; ∂ x ( h ( 0 ) + b ) = 0 ∂ x h ( 1 ) = 0 ∂ t z ( 0 ) = ∂ x ( h ( 0 ) u ( 0 ) ) ≡ ∂ x m ( 0 ) ∂ t m ( 0 ) + ∂ x ( h ( 0 ) ( u ( 0 ) ) 2 ) + h ( 0 ) ∂ x z ( 2 ) = 0 M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 7 / 49

Asymptotic expansion -rigorous analysis [Klainerman & Majda (’81), Feireisl & Novotn´ y (2009, 2013)] -formally: ( ε = Fr ) z ε ( x , t ; ε ) = z ( 0 ) ( x , t ) + ε z ( 1 ) ( x , t ) + ε 2 z ( 2 ) ( x , t ) u ε ( x , t ; ε ) = u ( 0 ) ( x , t ) + ε u ( 1 ) ( x , t ) + ε 2 u ( 2 ) ( x , t ) plug into the SWE = ⇒ z ( 0 ) = z ( 0 ) ( t ) ; ∂ x ( h ( 0 ) + b ) = 0 ∂ x h ( 1 ) = 0 ∂ t z ( 0 ) = ∂ x ( h ( 0 ) u ( 0 ) ) ≡ ∂ x m ( 0 ) ∂ t m ( 0 ) + ∂ x ( h ( 0 ) ( u ( 0 ) ) 2 ) + h ( 0 ) ∂ x z ( 2 ) = 0 limiting system as ε → 0 ( ∂ t b = 0 ) h ( 0 ) ( x ) = b ( x ) + const . (1) ∂ t h ( 0 ) = ∂ x m ( 0 ) ∂ t u ( 0 ) + u ( 0 ) ∂ x u ( 0 ) + ∂ x z ( 2 ) = 0 Does a numerical scheme give a consistent approximation of (1) ? M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 7 / 49

Time discretization Key idea: - semi-implicit time discretization: splitting into the linear and nonlinear part - linear operator modells gravitational (acoustic) waves are treated implicitly - rest nonlinear terms are treated explicitly M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 8 / 49

Time discretization Key idea: - semi-implicit time discretization: splitting into the linear and nonlinear part - linear operator modells gravitational (acoustic) waves are treated implicitly - rest nonlinear terms are treated explicitly ∂ w ∂ t = −∇ · F ( w ) + B ( w ) ≡ L ( w ) + N ( w ) w = ( z , m , n ) T , z = h + b ; b < 0 M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 8 / 49

Time discretization Key idea: - semi-implicit time discretization: splitting into the linear and nonlinear part - linear operator modells gravitational (acoustic) waves are treated implicitly - rest nonlinear terms are treated explicitly ∂ w ∂ t = −∇ · F ( w ) + B ( w ) ≡ L ( w ) + N ( w ) w = ( z , m , n ) T , z = h + b ; b < 0 − ∂ x ( m ) − ∂ y ( n )   b  Fr 2 ∂ x z  L ( w ) : =     b   Fr 2 ∂ y z M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 8 / 49

• L : spatially varying linear system w t + A 1 ( b ) w x + A 2 ( b ) w y = 0  0 1 0   0 0 1  − 1 ⇒ E L 0 0 0 A 1 = Fr 2 b ( x , y ) 0 0 A 2 = =     ∆ − 1 Fr 2 b ( x , y ) 0 0 0 0 0 Multi-d evolution operator in [Arun, M.L., Kraft, Prasad (2009)] M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 9 / 49

• L : spatially varying linear system w t + A 1 ( b ) w x + A 2 ( b ) w y = 0  0 1 0   0 0 1  − 1 ⇒ E L A 1 = Fr 2 b ( x , y ) A 2 = 0 0 0 = 0 0 ∆     − 1 Fr 2 b ( x , y ) 0 0 0 0 0 Multi-d evolution operator in [Arun, M.L., Kraft, Prasad (2009)] • REST: nonlinear system N z t = 0 1 m t + ( m 2 / ( z − b )) x + 2 Fr 2 ( z 2 ) x + ( mn / ( z − b ))) y = 0 1 n t + ( mn / ( z − b )) x + ( n 2 / ( z − b )) y + 2 Fr 2 ( z 2 ) y = 0 ⇒ E N = ∆ E N ∆ is the evolution along the characteristics or using an approximate Riemann solver M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 10 / 49

� P = ( x, y, t n + ∆ t ) t n t � i t Q 2 r c h s i t θ e Z � t Q 1 ( θ ) y x � n 1 ( θ ) Bicharacteristic scheme for linear operator L 2 M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 11 / 49

Wave propagation for the hyperbolic balance laws Information travels along bicharacteristic curves Integration along each curve + averaging over the cone mantle yields integral representation for the solution at the pick of the cone � P = ( x, y, t n + ∆ t ) t n t � t r i Q 2 h c s t θ i e Z � t Q 1 ( θ ) y x n 1 ( θ ) � M. Luk´ aˇ cov´ a-Medvid’ov´ a, K.W. Morton, and Gerald Warnecke. Finite volume evolution Galerkin methods for hyperbolic systems. J. Sci. Comp. 2004. M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 12 / 49

Back to our linear subsystem L ∂ t w − L ( w ) = 0 − div ( m , n ) T     z b  Fr 2 ∂ z / ∂ x  w : = m L ( w ) : =       n b   Fr 2 ∂ z / ∂ y M´ aria Luk´ aˇ cov´ a (Institute of Mathematics, Uni-Mainz) January, 2014 15 / 49