computational astrophysics 4 the godunov method · computational astrophysics 2009 romain teyssier...
TRANSCRIPT
![Page 1: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/1.jpg)
Computational Astrophysics 2009 Romain Teyssier
Computational Astrophysics 4
The Godunov method
Romain Teyssier
Oscar Agertz
![Page 2: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/2.jpg)
Computational Astrophysics 2009 Romain Teyssier
- Hyperbolic system of conservation laws
- Finite difference approximation
- The Modified Equation
- The Upwind scheme
- Von Neumann Analysis
- The Godunov Method
- Riemann solvers
- 2D Godunov schemes
Outline
![Page 3: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/3.jpg)
Romain TeyssierComputational Astrophysics 2009
HS of CL
System of conservation laws
- Vector of conservative variables
- Flux function
Integral form
x
t
x2x1
t1
t2
![Page 4: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/4.jpg)
Computational Astrophysics 2009 Romain Teyssier
Finite difference scheme
Finite difference approximation of the advection equation
![Page 5: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/5.jpg)
Computational Astrophysics 2009 Romain Teyssier
The Modified Equation
Taylor expansion in time up to second order
Taylor expansion in space up to second order
The advection equation becomes the advection-diffusion equation
Negative diffusion coefficient: the scheme is unconditionally unstable
![Page 6: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/6.jpg)
Romain TeyssierComputational Astrophysics 2009
The Upwind scheme
a>0: use only upwind values, discard downwind variables
Taylor expansion up to second order:
Upwind scheme is stable if C<1, with
![Page 7: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/7.jpg)
Romain TeyssierComputational Astrophysics 2009
Von Neumann analysis
Fourier transform the current solution:
Evaluate the amplification factor of the 2 schemes.
Fromm scheme:
Upwind scheme:
ω>1: the scheme is unconditionally unstable
ω<1 if C<1: the scheme is stable under the Courant condition.
![Page 8: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/8.jpg)
Romain TeyssierComputational Astrophysics 2009
The advection-diffusion equation
Finite difference approximation of the advection equation:
Central differencing unstable:
Upwind differencing is stable:
Smearing of initial discontinuity:
“numerical diffusion”
Thickness increases
as
![Page 9: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/9.jpg)
Romain TeyssierComputational Astrophysics 2009
The Godunov method
![Page 10: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/10.jpg)
Romain TeyssierComputational Astrophysics 2009
Finite volume scheme
Finite volume approximation of the advection equation:
Use integral form of the conservation law:
Exact evolution of volume averaged quantities:
Time averaged flux function:
![Page 11: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/11.jpg)
Romain TeyssierComputational Astrophysics 2009
Godunov scheme for the advection equation
The time averaged flux function:
is computed using the solution of the Riemann problem defined
at cell interfaces with piecewise constant initial data.
x
ui
ui+1
For all t>0:
The Godunov scheme for the advection equation is identical to the upwind finite difference scheme.
![Page 12: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/12.jpg)
Romain TeyssierComputational Astrophysics 2009
The time average flux function is computed using the self-similar solution of the inter-cell Riemann problem:
Godunov scheme for hyperbolic systemsThe system of conservation laws
is discretized using the following integral form:
This defines the Godunov flux:
Advection: 1 wave, Euler: 3 waves, MHD: 7 waves
Piecewise constant initial data
![Page 13: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/13.jpg)
Romain TeyssierComputational Astrophysics 2009
Riemann solvers
Exact Riemann solution is costly: involves Raphson-Newton iterations and complex non-linear functions.
Approximate Riemann solvers are more useful.
Two broad classes:
- Linear solvers
- HLL solvers
![Page 14: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/14.jpg)
Romain TeyssierComputational Astrophysics 2009
Linear Riemann solvers
Define a reference state as the arithmetic average or the Roe average
Evaluate the Jacobian matrix at this reference state.
Compute eigenvalues and (left and right) eigenvectors
Non-linear flux function with a linear diffusive term.
where
The interface state is obtained by combining all upwind waves
A simple example, the upwind Riemann solver:
![Page 15: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/15.jpg)
Computational Astrophysics 2009 Romain Teyssier
Riemann problem for adiabatic waves
Initial conditions are defined by 2 semi-infinite regions with piecewise constant initial states and .
x
t
Left state Right state
Left “star” state: (-,0,+)=(R,L,L) and right “star” state: (-,0,+)=(R,R,L).
2 mixed states
![Page 16: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/16.jpg)
Romain TeyssierComputational Astrophysics 2009
Approximate the true Riemann fan by 2 waves and 1 intermediate state:
HLL Riemann solver
x
t
UL UR
U*
Compute U* using the integral form between SLt and SRt
Compute F* using the integral form between SLt and 0.
![Page 17: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/17.jpg)
Romain TeyssierComputational Astrophysics 2009
Other HLL-type Riemann solvers
Lax-Friedrich Riemann solver:
HLLC Riemann solver: add a third wave for the contact (entropy) wave.
x
t
UL UR
U*L
SL SRS*
U*R
See Toro (1997) for details.
![Page 18: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/18.jpg)
Romain TeyssierComputational Astrophysics 2009
Sod test with the Godunov scheme
Lax-Friedrich Riemann solver
128 cells
![Page 19: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/19.jpg)
Romain TeyssierComputational Astrophysics 2009
Sod test with the Godunov scheme
HLLC Riemann solver
128 cells
![Page 20: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/20.jpg)
Romain TeyssierComputational Astrophysics 2009
Sod test with the Godunov scheme
Exact Riemann solver
128 cells
![Page 21: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/21.jpg)
Romain TeyssierComputational Astrophysics 2009
Multidimensional Godunov schemes
2D Euler equations in integral (conservative) form
Flux functions are now time and space average.
2D Riemann problems interact along cell edges:
Even at first order, self-similarity does not apply to the flux functions anymore.
Predictor-corrector schemes ?
![Page 22: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/22.jpg)
Romain TeyssierComputational Astrophysics 2009
Perform 1D Godunov scheme along each direction in sequence.
X step:
Y step:
Change direction at the next step using the same time step.Compute Δt, X step, Y step, t=t+Δt Y step, X step t=t+Δt
Courant factor per direction:
Courant condition:
Cost: 2 Riemann solves per time step.Second order based on corresponding 1D higher order method.
Directional (Strang) splitting
![Page 23: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/23.jpg)
Runge-Kutta schemePredictor step using the Godunov scheme and Δt/2.Flux functions computed using 1D Riemann problem at time tn+1/2 in each normal direction.4 Riemann solves per step.Courant condition:
Corner Transport UpwindPredictor step in transverse direction only using the 1D Godunov scheme.Flux functions computed using 1D Riemann problem at time tn+1/2 in each normal direction.4 Riemann solves per step.Courant condition:
Romain TeyssierComputational Astrophysics 2009
Unsplit schemes
Godunov schemeNo predictor step.Flux functions computed using 1D Riemann problem at time tn in each normal direction.2 Riemann solves per step.Courant condition:
![Page 24: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/24.jpg)
Romain TeyssierComputational Astrophysics 2009
The Godunov scheme for 2D advection
Perform a 2D unsplit conservative update
Solve 1D Riemann problem at each face
We get the following first-order linear scheme
Modified equation for 2D advection equation (exercise):
Differential form has 2 positive eigenvalues if:
and
![Page 25: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/25.jpg)
Romain TeyssierComputational Astrophysics 2009
CTU scheme for 2D advection
Solve 1D Riemann problem at each face using transverse predicted states
Predicted states are obtained in each direction by a 1D Godunov scheme.
during t/2
We get the following first-order linear scheme
Diffusion term in the modified equation is now (exercise):
![Page 26: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/26.jpg)
Romain TeyssierComputational Astrophysics 2009
Advection of a square with Godunov scheme
![Page 27: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/27.jpg)
Romain TeyssierComputational Astrophysics 2009
Advection of a square with Godunov scheme
![Page 28: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/28.jpg)
Romain TeyssierComputational Astrophysics 2009
Second-order Godunov scheme
![Page 29: Computational Astrophysics 4 The Godunov method · Computational Astrophysics 2009 Romain Teyssier Von Neumann analysis Fourier transform the current solution: Evaluate the amplification](https://reader034.vdocument.in/reader034/viewer/2022042222/5ec911df61588613f32911b1/html5/thumbnails/29.jpg)
Computational Astrophysics 2009 Romain Teyssier
Conclusion
- Upwind scheme for stability
- Modified equation analysis and numerical diffusion
- Godunov scheme: self-similarity of the Riemann solution
- Riemann solver: wave-by-wave upwinding
- Multiple dimensions: predictor-corrector scheme
- Need for higher-order schemes
Next lecture: Hydrodynamics 4