multi-moments moments or multi-moment moment constraints:

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1 Isaac Newton Institute for Mathematical Sciences Program on Multiscale Numerics for the Atmosphere and Ocean Cambridge, Nov. 16, 2012 High order finite volume methods using multi-moments moments or multi-moment moment constraints: basic idea, numerical formulations & applications to geophysical fluid dynamics F Xiao 1, CG Chen 2, S Ii 3, XL Li 4 1 Tokyo Institute of Technology 2 Xi an Jiaotong University 3 Osaka University 4 China Meteorological Administration

2 High order schemes with local reconstruction Adaptivity Unstructured mesh Multi-domain mesh What we expect: Spectral/geometric convergence Existing works Spectral element /hp FEM (Patera, Hughes, Karniadakis, ) Discontinuous Galerkin (Cockburn, Shu, Hesthaven, ) Spectral collocation/ Spectral volume / difference methods (Kopriva, Wang, ) CIP (Yabe ) Multi-moment based finite volume formulations (Since 2002)

3 Moment of Musical (Schubert In F minor) Quantities reflect spatial distribution of physical field

4 Outline Multi-moment finite volume method Lagrange interpolation & Hermite interpolation ti Multi-moment constrained collocation method / collocation method with multimoment constraints Schemes, analysis Applications in environmental and geophysical fluid dynamics Summary & future work

5 The first thought of using multi-moments for spatial discretization Mesh element X 1 X 8 X X 1 2 S 1 X 6 Ω i X Ω 2 X 7 Ω i X 3 S 4 Ω i S 2 X 5 X 4 X 3 X 4 X X S3 6 X 5 Moments Volume integrated average(via) Point value (PV) Surface integrated average (SIA) A multi-moment FVM memorizes and updates all of these. Never confuse a PV with a VIA in any possible way!

6 Flexible and efficient way to update the moments (Hyperbolic equation) 1 Point value (PV) not necessarily conservative Semi-Lagrangian approach (Multi-moment interpolation) Ω i Single-cell interpolation for advection and Mach cone Eulerian approach (Point-wise Riemann solver + RK) q + y + q x q x qy q + y + q x q q x y 2 Volume integrated average (VIA) conservative Finite volume formulation Ω i Single-cell interpolation for local DRPs

7 A multi-moment finite volume formulation of arbitrary order (1D) Predicted variables (unknowns) Evolution eqs. for unknowns Numerical formulation for spatial derivatives of flux function

8 Numerical derivative flux functions - Generalized Riemann problems Kolgan(70s); Ben-Artzi,Li, Falcovitz(84, 89,06,07.09); Aoki (1997), Toro et al. [ADER](02, ) Approximate solvers Linearization, reconstruction of state variables or characteristic variables Direct reconstruction of fluxes

9 Include slope limiter as an additional constraint Numerical test for scalar conservation law advected square wave (after 1000 steps)

10 Numerical results for 1D problems Convergence rate based on grid refinement (advection eq.) Two interacting blast waves (Euler eqs. 400 cells) Third-order Third-order Numerica al error Fourth-order Fourth-order Δx

11 Interaction of moving vortex and stationary shock Multi-moment FVM (3 rd -order) on unstructured mesh WENO (5 th -order) on structured mesh (Jiang & Shu, 96)

12 Some remarks on direct multi-moment moment formulation Perfect in 1D (accuracy, stability, efficiency, limiting) Practical multi-d scheme of 3 rd order accuracy can be built with point value and integrated moments (Integrated over line, surface, volume) Difficult to handle the differential moments in multi-d

13 Alternative approach: Combine Lagrange and Hermite Compute solution at the collocation points Construct the flux function with multi-moment moment constraints Multi-moment constrained collocation formulation Collocation method with multi-moment constraints

14 Outline Multi-moment finite volume method Lagrange interpolation & Hermite interpolation ti Multi-moment constrained collocation method / collocation method with multimoment constraints Schemes, analysis Applications in environmental and geophysical fluid dynamics Summary & future work

15 Lagrange form vs Hermite form (Non-cononical form)

16 Convergence of Hermite (multi-moment) interpolation 2 nd order 4 th order L2Error= L2Error= th order L2Error= e-1 8 th order L2Error= e-3 10 th order L2Error= e-4 Boyd, 2000

17 The Runge problem of Interpolation Polynomial of 10 degrees with equally-spaced points Pure Lagrange Pure Hermite Lagrange +Hermite(1) Lagrange +Hermite(2)

18 Outline Multi-moment finite volume method Lagrange interpolation & Hermite interpolation ti Multi-moment constrained collocation method / collocation method with multimoment constraints Schemes, analysis Applications in environmental and geophysical fluid dynamics Summary & future work

19 General form of collocation formulation of differential /nodal form Conservation law Mesh element Reconstructed cted flux function Riemann problem

20 Primary flux reconstruction

21 The tasks left to do 1 Find the ( derivative) fluxes at cell boundaries from the primary Lagrange interpolation 2 Construct the modified flux function using the available information from the primary Lagrange interpolation We are given more than we need!

22 Modified flux reconstruction of Huynh Lagrange formulation (1) Huynh, 2007

23 Modified flux reconstruction Lagrange formulation (2)

24 Numerical test of 3-point schemes based on single-moment/lagrange reconstruction 3-point nodal discontinuous Galerkin method 3-point collocation method on Guass-Legendre points Advection of a square pulse (1000steps)

25 Modified flux reconstruction using multi-momentmoment formulation

26 The general form of multi-moment constraint flux reconstruction (MMC-FR) Find the reconstructed flux function from both multimoment constraints and collocation constraints Solution points Constraint points Update the solutions in differential form

27 Outline Multi-moment finite volume method Lagrange interpolation & Hermite interpolation ti Multi-moment constrained collocation method / collocation method with multimoment constraints Schemes, analysis Applications in environmental and geophysical fluid dynamics Summary & future work

28 Example. Multi-moment constrained finite it volume method (MCV3) 3rd Local coordinate Multi-moment constraints Reconstructed flux function

29 Example. Multi-moment constrained finite it volume method (MCV3) 3rd MCV 3 rd (Equ-spaced solution points) Equations to update the nodal values Numerical conservation

30 Example. Multi-moment constrained finite it volume method (MCV3) 3rd MCV 3 rd (Chebyshev Gauss solution points) Equations to update the nodal values Numerical conservation

31 Comparison of SG, DG and MMC-FR (3-point schemes with Gauss points) Chebyshev collocation (Staggered grid) DG_Nodal MCV3 Advection of a square pulse (1000steps)

32 Location of solution points is not a sensitive matter flexibility in applications Equ-spacing points Gauss Chebyshev points P: -1/4,1/4,3/4 Advection of a square pulse (1000steps)

33 Flux reconstruction with multi-moment constraints at cell center Chebyshev points with center constraints: MCV3-CPCC Multi-moment constraints Chebyshev solution points Reconstructed flux function

34 MCV3 scheme for Chebyshev points with center constraints: t MCV3-CPCCCPCC Equations to update the nodal values Numerical conservation

35 Flux reconstruction with multi-moment constraints at cell center Uniform points with center constraints: MCV3-UPCC Multi-moment constraints Equi-spaced solution points Reconstructed flux function

36 MCV3 scheme for uniform points with center constraints: t MCV3-UPCC Discontinuous Interface value Discontinuous Interface value Equations to update the nodal values Numerical conservation

37 Fourier analysis Solution vector Amplification matrix

38 The Eigenvalue & Spectrum of Space-discretized Discontinuous Galerkin Spectral collocation MCV3 (equi-spaced points) Real part of the eigenvalues are negative (stable) The MCV3 schemes have smaller spectral radius and thus larger allowable CFL number for computational stability MCV3 (Gauss points) Stable region of 3 rd Runge Kutta

39 The spectra of the three-point schemes MCV3 MCV3_CPCC MCV3_UPCC Maximum allowable CFL No. MCV3 < MCV3_CPCC CPCC < MCV3_UPCC

40 Comparison of truncation errors Taylor expansion of the eigenvalues with respect to the grid spacing All computational modes are damped out at O(1/Δx) exponentially MCV3_UPCC and MCV3_CPCC are more accurate than the original MCV3 MCV3_CPCC is two-order more accurate in numerical dissipation, but has a different property in numerical dispersion

41 Comparison of dispersion and dissipation errors

42 Advection tests Advection of square and Gussain pulses (1000steps) MCV3 MCV3_UPCC MCV3_CPCC

43 4 th order multi-moment constrained finite it volume method MCV 4 th (Equ-spacing points) MCV 4 th (Gauss points) Advection of a square pulse (1000steps)

44 5 th order multi-moment constrained finite it volume method MCV 5 th (Equ-spacing points) MCV 5 th (Gauss points) Advection of a square pulse (1000steps)

45 5 th order scheme with point-value constraints at two inner points MCV5_pv24 (Equ-spacing points) (Gauss points) Advection of a square pulse (1000steps)

46 5 th order scheme with 2 nd -order derivatives constraints at two inner points MCV5_ 2D24 Advection of a square pulse (1000steps)

47 The spectra of spatial discretizations Spectral radius Maximum stable CFL number estimated from numerical experiments (3 rd RK time scheme)

48 Comparison of dispersion

49 Comparison of dissipation

50 Numerical results of Jiang-Shu advection test MCV3 MCV4_C2D MCV4 Constraints of the point values at internal points improve numerical accuracy, but tends to suffer a more restrictive CFL condition for computational stability. MCV5 MCV5_2D24 MCV5_PV24 Constraints in terms of the 2nd-order derivatives, i.e. the curvature of the primary reconstruction, greatly relieve CFL restriction for computational stability. Results after 1000steps

51 Validations for the 5 th order scheme Convergence rate of 5 th order scheme on refining grids (1D advection of density perturbation) Grid l_1 l_1 l_2 l_2 l_ infty l_ infty error order error order error order E E E E E E E E E E E E E E E

52 Validations (5 th -order scheme, Euler equation) Convergence rate on refining grids (2D density perturbation) Grid l_ 1 l_ 1 l_ 2 l_ 2 l _ infty l _ infty error order error order error order E E E E E E E E E E E E E E E Grid Convergence rate on refining grids (isentropic vortex) l_1 error l_1 order l_2 error l_2 order l_ infty error E E E-5 l_ infty order E E E E E E E E E

53 5 th order scheme with TVB limiter (100 cells) Sod s problem Lax s problem

54 5 th order scheme with TVB limiter it Shock-turbulence interaction (200 cells) Two interacting blast waves (400 cells)

55 5 th order scheme with TVB limiter Gid Grid: Gid Grid: Grid: Grid: Mach 3 wind tunnel with a step (M=5)

56 5 th order scheme with TVB limiter Grid: Grid: Grid: Grid: Double Mach Reflection (M=5)

57 Outline Multi-moment finite volume method Lagrange interpolation & Hermite interpolation ti Multi-moment constrained collocation method / collocation method with multimoment constraints Schemes, analysis, numerical tests Applications in environmental and geophysical fluid dynamics Summary & future work

58 Multi-function numerical model for multi-phase environmental fluid dynamics

59 Development of new generation global models Uniform spherical grids Complexity/difficulties: Overset, coordinate Lat/Lon grid discontinuity, Unstructured t configuration Triangular geodesic grid Hexagonal geodesic grid Gnomonic cubic grid Yin-Yang overset grid

60 Global models based on multi-moment moment (constrained ) finite volume method Triangular geodesic grid Hexagonal geodesic grid Gnomonic cubic grid Yin-Yang overset grid Williamson Benchmark test (case 6: Rossby-Haurwitz Wave)

61 Non-hydrostatic compressible model for atmosphere Join Dr. Chen s seminar on next Monday (Nov.19) U w for details

62 Summary & future work Using multi-moment and multi-moment constraints provides a general framework to construct t conservative, simple, flexible and efficient high-order schemes on local base. High order schemes can be more physically compatible and intuitive. A promising numerical platform for atmospheric/oceanic dynamic cores. New schemes with more attractive numerical features. More improvements and applications. Including physical constraints.

63 Some references F. Xiao, S. Ii, C.G. Chen and, X.L. Li, A note on the general multi-moment constrained flux reconstruction formulation for high order schemes, Appl. Math. Modell. (2012), /j.apm X.L.Li,, X. Shen, X.D. Peng, F. Xiao, Z.R. Zhuang and C.G. Chen, An accurate multi-moment constrained finite volume transport model on Yin-Yang grid, Advances in Atmospheric Sciences, submitted. X. Li, C. Chen, X. Shen and F. Xiao, A multi-moment constrained finite volume model for nonhydrostatic atmospheric dynamics, Mon. Wea. Rev., accepted. C.G. Chen, J.Z. Bin, F. Xiao, X.L. Li and X.S. Shen, A global Shallow water model on icosahedral- hexagonal grid by multi-moment constrained finite volume scheme, Quarterly Journal of the Royal Meteorological Society, in revision C.G.Chen, J.Z.Bin and F. Xiao: A Global Multimoment Constrained Finite-Volume Scheme for Advection Transport on the Hexagonal Geodesic Grid, Mon. Wea. Rev., 140, (2012) C.G.Chen, F. Xiao and X.L. Li: An adaptive multi-moment global model on cubed sphere, Mon. Wea. Rev. 139, (2011) C.G..Chen, F. Xiao, X.L. Li and Y.Yang: A multi-moment transport model on cubed-sphere grid, Int. J. Numer. Method in Fluids, 67, (2011) R.Akoh, S.Ii and F. Xiao: A multi-moment finite volume formulation for shallow water equations on unstructured mesh, J. Comput. Phys., 229, (2010) S.Ii and F. Xiao: A global shallow water model using high order multi-moment constrained finite volume method and icosahedral grid, J. Comput. Phys., 229, (2010) S.Ii and F. Xiao: High order multi-moment constrained finite volume method. Part I: Basic formulation, J. Comput. Phys., 228, (2009) C.G.Chen and F.Xiao: Shallow Water Model on Cubed-Sphere by Multi-moment Finite Volume Method. J. Comput. Phys. 227, (2008) X.L.Li, D.H.Chen, X.D. Peng, K. Takahashi and F. Xiao: A multi-moment finite volume shallow water model on Yin-Yang overset spherical grid, Mon. Wea. Rev. 136, (2008)

64 Some references (continued) R.Akoh, R h S. Ii and F. Xiao: A CIP/multi-moment t finiteit volume method for shallow water equations with source terms, Int. J. Numer. Method in Fluid, 56, (2008) S.Ii and F.Xiao: CIP/multi-moment finite volume method for Euler equations, a semi-lagrangian characteristic formulation, J. Comput. Phys., 222, (2007) F.Xiao, X.D. Peng and X.S. Shen: A finiteit volume grid using multi-moments t for geostrophic adjustment, Monthly Weather Review, 134, (2006) X.L.Li, D.H.Chen, X.D.Peng, F.Xiao and X.S.Chen: Implementation of the semi-lagrangian advection scheme on a quasi-uniform overset grid on a sphere, Advances in Atmospheric sciences, 23, (2006) X.D.Peng, F.Xiao and K.Takahashi: Global conservation constraint for a quasi-uniform overset grid on sphere, Quart. J. Roy. Meteor. Soc., 132, (2006) F.Xiao, R.Akoh and S.Ii: Unified formulation for compressible and incompressible flows by using multi integrated t moments II: multi-dimensional i l version for compressible and incompressibleibl flows, J. Comput. Phys., 213, (2006) S.Ii, M.Shimuta and F.Xiao: A 4th-order and single-cell-based advection scheme on unstructured grids using multi-moments, Comput. Phys. Commun., 173, (2005) F.Xiao, A.Ikebata and T.Hasegawa: Numerical simulations of free-interface f fluids by amulti integrated t moment method, Computers & Structures, 83, (2005) F.Xiao: Unified formulation for compressible and incompressible flows by using multi integrated moments I: One-dimensional inviscid compressible flow, J. Comput. Phys., 195, (2004) F.Xiao, T.Yabe, X.D.Peng and H.Kobayashi: Conservative and oscillation-less l atmospheric transportt schemes based on rational functions, J. Geophys. Res. 107 (D22), 4609, doi: /2001JD (2002)

65 Thank you!