Vorticity transport 2D using Lax-Wendroff 2 steps scheme
$begingroup$
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$zeta_t+u.zeta_x + v.zeta_y=frac{1}{text{Re}}(zeta_{xx}+zeta_{yy})$$
I have to use the Lax-Wendroff 2 steps scheme. I've made a discretization:
Lax-Wendroff 2 Steps:
1º STEP:
$$zeta^{n+1/2}_{i,j} = S^n_{12} - 2Omega(u_{i,j}Delta I^n_{i+1/2,j} + v_{i,j}Delta J^n_{i,j+1/2}) + Theta(S^n_{
32} - S^n_{12})$$
2º STEP:
$$zeta^{n+1}_{i,j} = zeta^{n}_{i,j} - Omega(u_{i,j}Delta I^{n+1/2}_{i+1/2,j} + v_{i,j}Delta J^{n+1/2}_{i,j+1/2}) + Theta(S^n - 4zeta^{n}_{i,j})$$
where,
$$S^n_{12} = zeta^n_{i+1/2,j} + zeta^n_{i-1/2,j} + zeta^n_{i,j+1/2} + zeta^n_{i,j-1/2}$$
$$S^n_{32} = zeta^n_{i+3/2,j} + zeta^n_{i-3/2,j} + zeta^n_{i,j+3/2} + zeta^n_{i,j-3/2}$$
$$Delta J^n_{i+1/2,j} = zeta^n_{i+1/2,j} - zeta^n_{i-1/2,j}$$
$$Delta J^n_{i+3/2,j} = zeta^n_{i,j+1/2} - zeta^n_{i,j-1/2}$$
$$Omega = frac{Delta t}{Delta x} $$
$$Theta = frac{Delta t}{Delta x^2,text{Re}} $$
First question: I'm not sure if this discretization is correct. Second problem: i don't know how to analyze the stability condition. Can i use the same stability condition for a Lax-Wendroff 1-step scheme?
Results: VORTICITY
pde numerical-methods fluid-dynamics transport-equation
edited Dec 8 '18 at 11:30
Harry49
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
$endgroup$
|
show 1 more comment
$begingroup$
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$zeta_t+u.zeta_x + v.zeta_y=frac{1}{text{Re}}(zeta_{xx}+zeta_{yy})$$
I have to use the Lax-Wendroff 2 steps scheme. I've made a discretization:
Lax-Wendroff 2 Steps:
1º STEP:
$$zeta^{n+1/2}_{i,j} = S^n_{12} - 2Omega(u_{i,j}Delta I^n_{i+1/2,j} + v_{i,j}Delta J^n_{i,j+1/2}) + Theta(S^n_{
32} - S^n_{12})$$
2º STEP:
$$zeta^{n+1}_{i,j} = zeta^{n}_{i,j} - Omega(u_{i,j}Delta I^{n+1/2}_{i+1/2,j} + v_{i,j}Delta J^{n+1/2}_{i,j+1/2}) + Theta(S^n - 4zeta^{n}_{i,j})$$
where,
$$S^n_{12} = zeta^n_{i+1/2,j} + zeta^n_{i-1/2,j} + zeta^n_{i,j+1/2} + zeta^n_{i,j-1/2}$$
$$S^n_{32} = zeta^n_{i+3/2,j} + zeta^n_{i-3/2,j} + zeta^n_{i,j+3/2} + zeta^n_{i,j-3/2}$$
$$Delta J^n_{i+1/2,j} = zeta^n_{i+1/2,j} - zeta^n_{i-1/2,j}$$
$$Delta J^n_{i+3/2,j} = zeta^n_{i,j+1/2} - zeta^n_{i,j-1/2}$$
$$Omega = frac{Delta t}{Delta x} $$
$$Theta = frac{Delta t}{Delta x^2,text{Re}} $$
First question: I'm not sure if this discretization is correct. Second problem: i don't know how to analyze the stability condition. Can i use the same stability condition for a Lax-Wendroff 1-step scheme?
Results: VORTICITY
pde numerical-methods fluid-dynamics transport-equation
edited Dec 8 '18 at 11:30
Harry49
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
$endgroup$
1
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
1
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31
|
show 1 more comment
$begingroup$
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$zeta_t+u.zeta_x + v.zeta_y=frac{1}{text{Re}}(zeta_{xx}+zeta_{yy})$$
I have to use the Lax-Wendroff 2 steps scheme. I've made a discretization:
Lax-Wendroff 2 Steps:
1º STEP:
$$zeta^{n+1/2}_{i,j} = S^n_{12} - 2Omega(u_{i,j}Delta I^n_{i+1/2,j} + v_{i,j}Delta J^n_{i,j+1/2}) + Theta(S^n_{
32} - S^n_{12})$$
2º STEP:
$$zeta^{n+1}_{i,j} = zeta^{n}_{i,j} - Omega(u_{i,j}Delta I^{n+1/2}_{i+1/2,j} + v_{i,j}Delta J^{n+1/2}_{i,j+1/2}) + Theta(S^n - 4zeta^{n}_{i,j})$$
where,
$$S^n_{12} = zeta^n_{i+1/2,j} + zeta^n_{i-1/2,j} + zeta^n_{i,j+1/2} + zeta^n_{i,j-1/2}$$
$$S^n_{32} = zeta^n_{i+3/2,j} + zeta^n_{i-3/2,j} + zeta^n_{i,j+3/2} + zeta^n_{i,j-3/2}$$
$$Delta J^n_{i+1/2,j} = zeta^n_{i+1/2,j} - zeta^n_{i-1/2,j}$$
$$Delta J^n_{i+3/2,j} = zeta^n_{i,j+1/2} - zeta^n_{i,j-1/2}$$
$$Omega = frac{Delta t}{Delta x} $$
$$Theta = frac{Delta t}{Delta x^2,text{Re}} $$
First question: I'm not sure if this discretization is correct. Second problem: i don't know how to analyze the stability condition. Can i use the same stability condition for a Lax-Wendroff 1-step scheme?
Results: VORTICITY
pde numerical-methods fluid-dynamics transport-equation
edited Dec 8 '18 at 11:30
Harry49
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
$endgroup$
I have to solve a 2D vorticity transport equation (very similar to viscous Burgers 2D). I had to change the equation for a non-dimensional form.
Vorticity transport Eq.:
$$zeta_t+u.zeta_x + v.zeta_y=frac{1}{text{Re}}(zeta_{xx}+zeta_{yy})$$
I have to use the Lax-Wendroff 2 steps scheme. I've made a discretization:
Lax-Wendroff 2 Steps:
1º STEP:
$$zeta^{n+1/2}_{i,j} = S^n_{12} - 2Omega(u_{i,j}Delta I^n_{i+1/2,j} + v_{i,j}Delta J^n_{i,j+1/2}) + Theta(S^n_{
32} - S^n_{12})$$
2º STEP:
$$zeta^{n+1}_{i,j} = zeta^{n}_{i,j} - Omega(u_{i,j}Delta I^{n+1/2}_{i+1/2,j} + v_{i,j}Delta J^{n+1/2}_{i,j+1/2}) + Theta(S^n - 4zeta^{n}_{i,j})$$
where,
$$S^n_{12} = zeta^n_{i+1/2,j} + zeta^n_{i-1/2,j} + zeta^n_{i,j+1/2} + zeta^n_{i,j-1/2}$$
$$S^n_{32} = zeta^n_{i+3/2,j} + zeta^n_{i-3/2,j} + zeta^n_{i,j+3/2} + zeta^n_{i,j-3/2}$$
$$Delta J^n_{i+1/2,j} = zeta^n_{i+1/2,j} - zeta^n_{i-1/2,j}$$
$$Delta J^n_{i+3/2,j} = zeta^n_{i,j+1/2} - zeta^n_{i,j-1/2}$$
$$Omega = frac{Delta t}{Delta x} $$
$$Theta = frac{Delta t}{Delta x^2,text{Re}} $$
First question: I'm not sure if this discretization is correct. Second problem: i don't know how to analyze the stability condition. Can i use the same stability condition for a Lax-Wendroff 1-step scheme?
Results: VORTICITY
pde numerical-methods fluid-dynamics transport-equation
pde numerical-methods fluid-dynamics transport-equation
edited Dec 8 '18 at 11:30
Harry49
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
edited Dec 8 '18 at 11:30
Harry49
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
edited Dec 8 '18 at 11:30
Harry49
6,40231132
edited Dec 8 '18 at 11:30
Harry49
6,40231132
edited Dec 8 '18 at 11:30
Harry49
6,40231132
6,40231132
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
asked Nov 30 '18 at 22:36
fvalencarfvalencar
114
114
1
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
1
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31
|
show 1 more comment
1
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
1
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31
1
1
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
1
1
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31
|
show 1 more comment
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1
$begingroup$
It's the convection-diffusion equation, isn't it?
$endgroup$
– rafa11111
Nov 30 '18 at 22:40
$begingroup$
Yes! The term with $zeta_x$ is the convection, and $zeta_{xx}$ is the diffusion term.
$endgroup$
– fvalencar
Nov 30 '18 at 22:46
$begingroup$
That's the first time I see this equation being called 'Viscous two-dimensional Burger's equation'. I'm not well acquainted with the Lax-Wendroff method, but your discretization seems to be alright. You will probably find the stability criteria in specialized literature, such as Fletcher's Computational Methods for Fluid Dynamics. However, I think it's much more common to solve this equation numerically with the FTCS scheme, and I have been solving this very equation using FTCS; if you consider the FTCS, you can find the stability criteria here.
$endgroup$
– rafa11111
Nov 30 '18 at 22:57
$begingroup$
@rafa11111 The name is vorticity transport equation non-dimensional, but it's very similar with viscous burger's equation. I've already solved this problem using the FTCS scheme, but i have to solve it with LW 2 Steps scheme too.
$endgroup$
– fvalencar
Nov 30 '18 at 23:03
1
$begingroup$
Unless $u$ and $v$ defined previously, you'll need some more equations.
$endgroup$
– Mattos
Nov 30 '18 at 23:31