Adaptive Grids in General Coordinates

Transcription

1 for Conservation Laws SE Numerische Methoden der Astrophysik Harald Höller

2 Astrophysical Motivation Numerical and Mathematical Motivation 1 Motivation Astrophysical Motivation Numerical and Mathematical Motivation 2 Hyperbolic Conservation Laws Mathematical Background 3 Nonlinear Differential Operators Conservation Laws 4 Methods The Orthogonal Option Hybrid Method

4 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Figure: Betelgeuse, Source (Link)

5 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Rotation-induced mixing Figure: Betelgeuse, Source (Link)

6 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Rotation-induced mixing Coupled pulsation modes Figure: Betelgeuse, Source (Link)

7 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Rotation-induced mixing Coupled pulsation modes Convection Figure: Betelgeuse, Source (Link)

8 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Rotation-induced mixing Coupled pulsation modes Convection Accretion discs Figure: Betelgeuse, Source (Link)

9 Astrophysical Motivation Numerical and Mathematical Motivation Non-Spherically Symmetric Astrophysical Problems Spherical symmetry is a major constriction and comes into conflict with... Flattening by rotation Rotation-induced mixing Coupled pulsation modes Convection Accretion discs (Galactic) winds... Figure: Betelgeuse, Source (Link)

11 The Numerical Approach Astrophysical Motivation Numerical and Mathematical Motivation RHD numerics - the status quo Conventional stellar evolution codes are 1D - spherically symmetric Figure: 3D Convection, H. Muthsam, Source (Link)

12 Astrophysical Motivation Numerical and Mathematical Motivation The Numerical Approach RHD numerics - the status quo Conventional stellar evolution codes are 1D - spherically symmetric 2D and 3D RHD Codes often emphasize on temporally and spatially small scales (convection) Figure: 3D Convection, H. Muthsam, Source (Link)

13 Astrophysical Motivation Numerical and Mathematical Motivation The Numerical Approach RHD numerics - the status quo Conventional stellar evolution codes are 1D - spherically symmetric 2D and 3D RHD Codes often emphasize on temporally and spatially small scales (convection) Explicit codes are parallelizable but time steps are limited by CFL-condition Figure: 3D Convection, H. Muthsam, Source (Link)

14 Astrophysical Motivation Numerical and Mathematical Motivation The Numerical Approach RHD numerics - the status quo Conventional stellar evolution codes are 1D - spherically symmetric 2D and 3D RHD Codes often emphasize on temporally and spatially small scales (convection) Explicit codes are parallelizable but time steps are limited by CFL-condition Special non Euclidean geometries demand problem-oriented grids Figure: 3D Convection, H. Muthsam, Source (Link)

15 The Mathematical Challenge Astrophysical Motivation Numerical and Mathematical Motivation Mathematical treatment of RHD numerics Numerical methods for nonlinear conservation laws Figure: Slightly non-orthogonal polar grid

16 Astrophysical Motivation Numerical and Mathematical Motivation The Mathematical Challenge Mathematical treatment of RHD numerics Numerical methods for nonlinear conservation laws Weak, non differentiable Shock-solutions have to be considered Figure: Slightly non-orthogonal polar grid

17 Astrophysical Motivation Numerical and Mathematical Motivation The Mathematical Challenge Mathematical treatment of RHD numerics Numerical methods for nonlinear conservation laws Weak, non differentiable Shock-solutions have to be considered Nonlinear coordinate systems pose a extra effort to preliminary tensor analysis Figure: Slightly non-orthogonal polar grid

18 Hyperbolic Conservation Laws Mathematical Background 1 Motivation Astrophysical Motivation Numerical and Mathematical Motivation 2 Hyperbolic Conservation Laws Mathematical Background 3 Nonlinear Differential Operators Conservation Laws 4 Methods The Orthogonal Option Hybrid Method

19 Hyperbolic Conservation Laws Mathematical Background Basic Definitions Density and flux functions Mathematically the equations of radiation hydrodynamics form a system of hyperbolic conservation laws that describe the interaction of a density function d and its flux f d : R n [0, ) R m f : U R m, U R m, U open.

20 Basic Definitions Motivation Hyperbolic Conservation Laws Mathematical Background Density and flux functions Mathematically the equations of radiation hydrodynamics form a system of hyperbolic conservation laws that describe the interaction of a density function d and its flux f d : R n [0, ) R m f : U R m, U R m, U open. The temporal and spatial change of the integrated density in a connected space Ω R n then equals the flux over the boundary Ω, where S is the outward oriented surface element. t ddv + f ds = 0 t > 0 (1) Ω Ω

21 Basic Definitions Motivation Hyperbolic Conservation Laws Mathematical Background RHD conservation laws For the in physical problem of radiation hydrodynamics this system gains the following form. ( ρ(x, ) t) ρu ρu (x,t) ( ) d(x,t) = ρǫ (x,t) J γ (x,t;n,ν), f(d) = uρu T + P ( ) ρǫ + P u (2) cf γ F γ (x,t;n,ν) c 2 P γ

22 Basic Definitions Motivation Hyperbolic Conservation Laws Mathematical Background RHD conservation laws For the in physical problem of radiation hydrodynamics this system gains the following form. ( ρ(x, ) t) ρu ρu (x,t) ( ) d(x,t) = ρǫ (x,t) J γ (x,t;n,ν), f(d) = uρu T + P ( ) ρǫ + P u (2) cf γ F γ (x,t;n,ν) c 2 P γ With J γ the mean spectral intensity and its frist two momenta F γ (x,t;ν) = I γ ndω, P γ (x,t;ν) = 1 I γ nn T dω c

23 Theory of Conservation Laws Hyperbolic Conservation Laws Mathematical Background The mathematical framework provides... Theory of weak solutions

24 Hyperbolic Conservation Laws Mathematical Background Theory of Conservation Laws The mathematical framework provides... Theory of weak solutions Closure relation for equations of radiative transfer

25 Hyperbolic Conservation Laws Mathematical Background Theory of Conservation Laws The mathematical framework provides... Theory of weak solutions Closure relation for equations of radiative transfer Mathematical motivation for numerical specialities like artificial viscosity

26 Hyperbolic Conservation Laws Mathematical Background Theory of Conservation Laws The mathematical framework provides... Theory of weak solutions Closure relation for equations of radiative transfer Mathematical motivation for numerical specialities like artificial viscosity Conservative differential operators Γ g... [ g... ê... ] (3) References e.g. Randall J. LeVeque (1990) [1], Thompson, Warsi, Mastin (1985) [2]

27 Nonlinear Differential Operators Conservation Laws 1 Motivation Astrophysical Motivation Numerical and Mathematical Motivation 2 Hyperbolic Conservation Laws Mathematical Background 3 Nonlinear Differential Operators Conservation Laws 4 Methods The Orthogonal Option Hybrid Method

28 Nonlinear Differential Operators Conservation Laws Notation and conventions Einstein notation is used - summation over upper and lower indices.

29 Nonlinear Differential Operators Conservation Laws Notation and conventions Einstein notation is used - summation over upper and lower indices. With boldface we denote tensorial quantities. x x = x µ e µ ê µ e µ

30 Nonlinear Differential Operators Conservation Laws Notation and conventions Einstein notation is used - summation over upper and lower indices. With boldface we denote tensorial quantities. x x = x µ e µ ê µ e µ Base vectors with hat (ê µ ) are not normalized.

31 Nonlinear Differential Operators Conservation Laws Notation and conventions Einstein notation is used - summation over upper and lower indices. With boldface we denote tensorial quantities. x x = x µ e µ ê µ e µ Base vectors with hat (ê µ ) are not normalized. Quantities with upper indices are called contravariant, quantities with lower indices covariant. Vector x is expressed by its contravariant components x µ with reference to its covariant base ê µ.

32 Nonlinear Differential Operators Conservation Laws Linear coordinate systems Coordinate lines of a linear coordinate system are straight lines on an affine space.

33 Nonlinear Differential Operators Conservation Laws Linear coordinate systems Coordinate lines of a linear coordinate system are straight lines on an affine space. Let M be an affine space with corresponding vector space V and Σ (i) a linear coordinate system. The coordinate system maps points x M (linear, bijective) to a d-tuple of numbers x µ. Σ (i) : M R d, x (x (i) µ ) (4)

34 Nonlinear Differential Operators Conservation Laws Linear coordinate systems Coordinate lines of a linear coordinate system are straight lines on an affine space. Let M be an affine space with corresponding vector space V and Σ (i) a linear coordinate system. The coordinate system maps points x M (linear, bijective) to a d-tuple of numbers x µ. Σ (i) : M R d, x (x (i) µ ) (4) Two linear coordinate systems are connected via x (i) µ = Λ (ij) µ ν x (j) ν + λ (ij) µ. (5)

35 Nonlinear Differential Operators Conservation Laws Base vectors We regard the coordinate lines as curves in our vector space and write down the common definition of the base vectors. ê (i)µ = x x µ (i) (6) The base vectors are the tangential vectors to the coordinate lines; the dual base is implicitly defined by ê µ (i) ê (i)ν = δ µ ν.

36 Nonlinear Differential Operators Conservation Laws Base vectors We regard the coordinate lines as curves in our vector space and write down the common definition of the base vectors. ê (i)µ = x x µ (i) (6) The base vectors are the tangential vectors to the coordinate lines; the dual base is implicitly defined by ê µ (i) ê (i)ν = δ µ ν. They transform unequally for lower and upper indices. ν ê (j)µ = Λ (ij) µ ê(i)ν = x ν (i) ê µ (j) = Λ (ij) µ ν êν (i) = x (i) µ x (j) µ ê(i)ν x (j) ν êν (i) (7)

37 Nonlinear Differential Operators Conservation Laws Linear Transformation Figure: Linear Transformation

38 Nonlinear Differential Operators Conservation Laws Higher rank tensors A Tensor T of rank (m,n) is a multilinear map T : V V }{{ V V R }}{{} m n which is expressed by a pattern of d m+n real numbers in coordinate system Σ (i) T(x) = T (i) µ...ν ρ...σêµ ê ν ê ρ ê σ (8) The entries T (i) are the components of T with respect to Σ (i).

39 Nonlinear Differential Operators Conservation Laws Higher rank tensors A Tensor T of rank (m,n) is a multilinear map T : V V }{{ V V R }}{{} m n which is expressed by a pattern of d m+n real numbers in coordinate system Σ (i) T(x) = T (i) µ...ν ρ...σêµ ê ν ê ρ ê σ (8) The entries T (i) are the components of T with respect to Σ (i). The transformation rule for such a tensor yields µ...ν T (j) = Λ ρ...σ (ji) µ...λ α (ji) ν Λ β (ij) l...λ ρ (ij) δ T σ (i) α...β l...δ (9)

41 Nonlinear Differential Operators Conservation Laws Nonlinear Nonlinear coordinate systems and local bases With nonlinear coordinate systems, the base vectors are no longer constant but functions of position. ê (i)µ (x) = x x (i) µ (10)

42 Nonlinear Differential Operators Conservation Laws Nonlinear Nonlinear coordinate systems and local bases With nonlinear coordinate systems, the base vectors are no longer constant but functions of position. ê (i)µ (x) = x x (i) µ (10) Of course, the transformation matrices Λ are now functions too Λ (ij) µ and pointwise following is valid (x) = x µ (i) (11) ν x ν (j) ê µ (i) (x) ê (i)ν(x) = δ µ ν. (12)

43 Nonlinear Differential Operators Conservation Laws Nonlinear Transformation Figure: Nonlinear Transformation

44 Nonlinear Differential Operators Conservation Laws Nonlinear Tensor fields The density and flux functions of our conservation laws are tensor fields; tensor fields are only defined with respect to a local base.

45 Nonlinear Differential Operators Conservation Laws Nonlinear Tensor fields The density and flux functions of our conservation laws are tensor fields; tensor fields are only defined with respect to a local base. T (i) µ...ν ρ...σ (x) = T(êµ (i) (x),...,êν (i) (x),ê (i)ρ(x),...ê (i)ρ (x)) (13)

46 Nonlinear Differential Operators Conservation Laws Nonlinear Tensor fields The density and flux functions of our conservation laws are tensor fields; tensor fields are only defined with respect to a local base. T (i) µ...ν ρ...σ (x) = T(êµ (i) (x),...,êν (i) (x),ê (i)ρ(x),...ê (i)ρ (x)) (13) All operations with tensor fields (addition, multiplication etc.) have to be carried out pointwise.

48 Nonlinear Differential Operators Conservation Laws Covariant Derivative Why we need a covariant derivative When is a tensor constant? T µ...ν ρ...σ ξ α =...? (14)

49 Nonlinear Differential Operators Conservation Laws Covariant Derivative Why we need a covariant derivative When is a tensor constant? T µ...ν ρ...σ ξ α =...? (14) Covariant derivative ν T µ = ν T µ + Γ µ ρνt ρ. (15)

50 Nonlinear Differential Operators Conservation Laws Covariant Derivative Why we need a covariant derivative When is a tensor constant? T µ...ν ρ...σ ξ α =...? (14) Covariant derivative ν T µ = ν T µ + Γ µ ρνt ρ. (15) Christoffel Symbols Γ µ ê ν = Γ ρ µνê ρ. (16)

51 Nonlinear Differential Operators Conservation Laws Metric Tensor Local measure of uniformity Metric tensor g

52 Nonlinear Differential Operators Conservation Laws Metric Tensor Local measure of uniformity Metric tensor g Connection between co- and contravariant components v µ = g µν v ν bzw. v µ = g µν v ν (17) where g µν (x)g νρ (x) = δ µ ρ (18)

53 Nonlinear Differential Operators Conservation Laws Metric Tensor Local measure of uniformity Metric tensor g Connection between co- and contravariant components v µ = g µν v ν bzw. v µ = g µν v ν (17) where g µν (x)g νρ (x) = δ µ ρ (18) Local base ê µ (x) ê ν (x) = g µν (x) (19)

55 Nonlinear Differential Operators Conservation Laws Conservative Formulation Numerical fluxes The conservation law t Ω ddv + Ω f ds = 0 (20)

56 Nonlinear Differential Operators Conservation Laws Conservative Formulation Numerical fluxes The conservation law t Ω ddv + Ω f ds = 0 (20)... has to be satisfied numerically by calculating the fluxes adequately δ(d V) δt }{{} density change + F(D) = 0 (21) }{{} fluxes

57 Nonlinear Differential Operators Conservation Laws Advection of Constant Density ρ Figure: Linear coordinate system

58 Nonlinear Differential Operators Conservation Laws Advection of Constant Density ρ Figure: Nonlinear coordinate system

59 Nonlinear Differential Operators Conservation Laws Conservative Formulation Analytical VS Numerical Analytically left and right hand side of (uρ) = u ρ + ρ u are equivalent. However, numerically we need to conserve the entries of the density function, like (uρ)

60 Nonlinear Differential Operators Conservation Laws Conservative Formulation Analytical VS Numerical Analytically left and right hand side of (uρ) = u ρ + ρ u are equivalent. However, numerically we need to conserve the entries of the density function, like (uρ) Components of tensors are not conserved; Christoffel symbols are unwelcome geometric source terms

61 Nonlinear Differential Operators Conservation Laws Conservative Formulation Analytical VS Numerical Analytically left and right hand side of (uρ) = u ρ + ρ u are equivalent. However, numerically we need to conserve the entries of the density function, like (uρ) Components of tensors are not conserved; Christoffel symbols are unwelcome geometric source terms Conservative covariant derivatives are motivated by the theory of differential forms and yield [2] e.g. gradφ = ê µ µ φ = ê µ µ φ = 1 g µ [ g φê µ ] (22)

62 Nonlinear Differential Operators Conservation Laws Conservative Formulation Conservative advection Typically advections terms look like K = A t + (ua) (23)

63 Nonlinear Differential Operators Conservation Laws Conservative Formulation Conservative advection Typically advections terms look like K = A t + (ua) (23) With conservative notation they yield [ ] [ ] g K = t g A µ g U µ A (24) with effective velocity U µ ê µ (u ẋ), where ẋ is the grid velocity

64 Nonlinear Differential Operators Conservation Laws Conservative Formulation Conservative advection Typically advections terms look like K = A t + (ua) (23) With conservative notation they yield [ ] [ ] g K = t g A µ g U µ A (24) with effective velocity U µ ê µ (u ẋ), where ẋ is the grid velocity Volume element g and the covariant base vectors emerge in each derivation

65 Methods The Orthogonal Option Hybrid Method 1 Motivation Astrophysical Motivation Numerical and Mathematical Motivation 2 Hyperbolic Conservation Laws Mathematical Background 3 Nonlinear Differential Operators Conservation Laws 4 Methods The Orthogonal Option Hybrid Method

66 Methods The Orthogonal Option Hybrid Method Implicit RHD on adaptive grids Ideally, shocks propagate with coordinate surfaces in order to avoid skew fluxes Figure: Skew shock

67 Methods The Orthogonal Option Hybrid Method Implicit RHD on adaptive grids Ideally, shocks propagate with coordinate surfaces in order to avoid skew fluxes Adaptive mesh refinement VS multi-dimensionally adaptive grids Figure: Skew shock

68 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 1: The grid should be adaptive in more than one dimension but remain orthogonal throughout the computation.

69 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 1: The grid should be adaptive in more than one dimension but remain orthogonal throughout the computation. The metric tensor would remain diagonal

70 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 1: The grid should be adaptive in more than one dimension but remain orthogonal throughout the computation. The metric tensor would remain diagonal E.g. tensor of artificial viscosity Q ij = Q lm g li g mj = (µ 1 + µ 2 max( divu,0)) (g li g mj ( l u m + j u m ) g ij 13 ) divu

71 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 1: The grid should be adaptive in more than one dimension but remain orthogonal throughout the computation. The metric tensor would remain diagonal E.g. tensor of artificial viscosity Q ij = Q lm g li g mj = (µ 1 + µ 2 max( divu,0)) (g li g mj ( l u m + j u m ) g ij 13 ) divu 27 2 compared to 3 2 (g ij = 0 i j) arithmetic operations at each node and time step

72 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 2: Grid cells should basically maintain their local geometries.

73 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 2: Grid cells should basically maintain their local geometries. Even if orthogonality can not be maintained, the adaptive grids needs some kind of mollifying property

74 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 2: Grid cells should basically maintain their local geometries. Even if orthogonality can not be maintained, the adaptive grids needs some kind of mollifying property Avoid excessively twisted, acute-angled or one another overtaking cells

75 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 2: Grid cells should basically maintain their local geometries. Even if orthogonality can not be maintained, the adaptive grids needs some kind of mollifying property Avoid excessively twisted, acute-angled or one another overtaking cells In terms of the metric tensor, the off diagonal elements should remain small {g ij i = j} {g ij i j}

76 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 3: The more dimensional grid should contain the one dimensional special case of radial geometry.

77 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 3: The more dimensional grid should contain the one dimensional special case of radial geometry. Compare calculations easily with 1D simulations

78 Methods The Orthogonal Option Hybrid Method The ideal grid should fulfill following criteria 3: The more dimensional grid should contain the one dimensional special case of radial geometry. Compare calculations easily with 1D simulations Non-orthogonality governed by a number of parameters polar grid, if they get turned off x(ξ,η) = (a 1 ξ + a 2 ξ 2 )(1 + α 1 η + α 2 η 2 + α 3 η 3 )cos η y(ξ,η) = (b 1 ξ + b 2 ξ 2 )(1 + β 1 η + β 2 η 2 + β 3 η 3 )sin η

80 Methods The Orthogonal Option Hybrid Method Equidistribution Uniform error distribution throughout the domain Local adjustments of mesh size - the steeper the gradient, the higher the spatial resolution

81 Methods The Orthogonal Option Hybrid Method Equidistribution Uniform error distribution throughout the domain Local adjustments of mesh size - the steeper the gradient, the higher the spatial resolution 1D equidistribution given via arc length

82 Methods The Orthogonal Option Hybrid Method Equidistribution Uniform error distribution throughout the domain Local adjustments of mesh size - the steeper the gradient, the higher the spatial resolution 1D equidistribution given via arc length 2D no canonical way, but metric tensor will play a major role

83 Methods The Orthogonal Option Hybrid Method Equidistribution Uniform error distribution throughout the domain Local adjustments of mesh size - the steeper the gradient, the higher the spatial resolution 1D equidistribution given via arc length 2D no canonical way, but metric tensor will play a major role Equidistribution in more than 1 dimension can only be satisfied locally References e.g. Dorfi & Drury (1987) [3] or Hunag & Sloan (1995) [4], Anderson (1987) [5]

84 Methods The Orthogonal Option Hybrid Method Differential Methods Example: Laplace System Differential equation techniques are popular with complex geometries

85 Methods The Orthogonal Option Hybrid Method Differential Methods Example: Laplace System Differential equation techniques are popular with complex geometries Properties of PDEs are used to control the coordinates and help to find solution methods

86 Methods The Orthogonal Option Hybrid Method Differential Methods Example: Laplace System Differential equation techniques are popular with complex geometries Properties of PDEs are used to control the coordinates and help to find solution methods Elliptical equations ensure smooth solutions, even if boundaries are non-smooth

87 Methods The Orthogonal Option Hybrid Method Differential Methods Example: Laplace System Differential equation techniques are popular with complex geometries Properties of PDEs are used to control the coordinates and help to find solution methods Elliptical equations ensure smooth solutions, even if boundaries are non-smooth E.g. with x(ξ) (x (α) i (ξ)) the Laplace system yields g lm 2 x i a coupled quasilinear system of PDEs. ξ l = 0, (25) ξm References e.g. Vladimir D. Liseikin (1999) [6], Vladimir D. Liseikin (2004) [7]

88 Methods The Orthogonal Option Hybrid Method Differential Methods Example: Laplace System Laplace and Poisson system are boundary value problems Grid concentration near a concave, grid rarefaction near convex boundary Figure: Nonlinear coordinate system

90 Methods The Orthogonal Option Hybrid Method Orthogonal Coordinate Transformation If orthogonality is a must-have... (Unfortunately) it can be shown, that the only orthogonal quasi-polar grid is polar; product ansatz x(ξ,η) = a(ξ)α(η), y(ξ,η) = b(ξ)β(η)

91 Methods The Orthogonal Option Hybrid Method Orthogonal Coordinate Transformation If orthogonality is a must-have... (Unfortunately) it can be shown, that the only orthogonal quasi-polar grid is polar; product ansatz x(ξ,η) = a(ξ)α(η), y(ξ,η) = b(ξ)β(η) Orthogonality condition 2 equations g 12 = g 21 = aαa α + bβb β = 0 (26) a 2 = λb 2 + const 1 β 2 = λα 2 + const 2, λ 0 (27)

92 Methods The Orthogonal Option Hybrid Method Orthogonal Coordinate Transformation If orthogonality is a must-have... With boundary conditions this yields a = λb, β = λ λα 2. λ is just a scaling parameter and can be set λ = 1 which yields a = b β = 1 α 2 (28) which leads to a = b = ξ and α = cos η, β = sin η

93 Methods The Orthogonal Option Hybrid Method Orthogonal Coordinate Transformation If orthogonality is a must-have... With boundary conditions this yields a = λb, β = λ λα 2. λ is just a scaling parameter and can be set λ = 1 which yields a = b β = 1 α 2 (28) which leads to a = b = ξ and α = cos η, β = sin η The only adaptive grid in 2D that remains orthogonal is strictly polar

94 Methods The Orthogonal Option Hybrid Method Orthogonal Coordinate Transformation If orthogonality is a must-have... With boundary conditions this yields a = λb, β = λ λα 2. λ is just a scaling parameter and can be set λ = 1 which yields a = b β = 1 α 2 (28) which leads to a = b = ξ and α = cos η, β = sin η The only adaptive grid in 2D that remains orthogonal is strictly polar Adaptive mesh refinement

95 Methods The Orthogonal Option Hybrid Method Mesh Refinement Refining in r Figure: Nonlinear coordinate system

96 Methods The Orthogonal Option Hybrid Method Mesh Refinement Refining in ϑ Figure: Nonlinear coordinate system

97 Methods The Orthogonal Option Hybrid Method : Example of Use in Mathematica Mathematica-File as pdf-document in the appendix

98 Methods The Orthogonal Option Hybrid Method Bibliography I R. J. LeVeque, Lectures in Mathematics, ETH-Zurich (1990). J. F. Thompson, Z. U. Warsi, and C. W. Mastin, Numerical grid generation: foundations and applications (Elsevier North-Holland, Inc., New York, NY, USA, 1985). E. Dorfi and L. Drury, Journal of Computational Physics 69, 175 (1987). W. Huang and D. M. Sloan, SIAM Journal on Scientific Computing 15, 776 (1994). D. A. Anderson, Applied Mathematics and Computation 24, 211 (1987).

99 Methods The Orthogonal Option Hybrid Method Bibliography II V. D. Liseikin, A Computational Differential Geometry Approach to (Springer-Verlag, Berlin Heidelberg, 2004). V. D. Liseikin, Methods (Springer-Verlag, Berlin Heidelberg, 1999).