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Adding artificial dissipation to continuity equation
In the numerical solution of the Wave Equation, using finite differences, where do I obtain the spatial values from?Diffusion of a chemical species inside a Y-shaped tubeNumerical solution of non-constant coefficient diffusion equation via finite-difference methodA posteriori measures of numerical dissipation and dispersionHow do I determine if the equation is a conservation law?Need Help Discretizing this Heat EquationWhat makes a numerical scheme conservative?Write the conservation equation for the heat equationInstability in 2D steady state heat equation with variable thermal diffusivityHow to numerically solve a partial differential equation containing two time derivatives?
$begingroup$
I'm trying to solve the system of equations that relates to blood flows in arteries i.e. flow in elastic tubes.
The system looks as follows
$$fracpartial Apartial t+fracpartialleft(Auright)partial x=0$$
$$ fracpartial upartial t+ufracpartial upartial x+frac1rhofracpartial ppartial x=0$$
The first equation is for volume conservation. Fluid velocity is $u$. Pressure $p$ and area $A$ are related directly by a "tube law" so we don't solve for $p$ and the only quantities that are solved for every timestep are $A$ and $u$.
The tube law has a dissipative term $partial A/partial t$ so through $partial p/partial x$ and the continuity equation, we end up with a $partial^2 u /partial x^2$ term so there's some nice dissipation for the momentum equation.
Now I wanted to quickly solve this using MATLAB's PDEPE and realized that I needed a second spatial derivative for the volume conservation equation as well so I modified it to be
$$fracpartial Apartial t+fracpartialleft(Auright)partial x + epsilonfracpartial^2 Apartial x^2=0$$
where $epsilon$ is really small.
I ran a couple of cases and am seeing nice physically sensible tube deformations and flows. I compute the residual for the original unadulterated volume conservation equation and the values seem acceptable.
But from a theoretical standpoint, I wanted to know how egregious this error is or if it is an acceptable hack for solving the system? I have converted a PDE that had first order spatial derivatives into one that has second order derivatives. Is the modified system completely different or is it the same as the original as $epsilon to 0$? Thanks!
pde numerical-methods systems-of-equations computational-mathematics finite-differences
asked Mar 12 at 21:01
shk92shk92
283
$endgroup$
add a comment |
$begingroup$
I'm trying to solve the system of equations that relates to blood flows in arteries i.e. flow in elastic tubes.
The system looks as follows
$$fracpartial Apartial t+fracpartialleft(Auright)partial x=0$$
$$ fracpartial upartial t+ufracpartial upartial x+frac1rhofracpartial ppartial x=0$$
The first equation is for volume conservation. Fluid velocity is $u$. Pressure $p$ and area $A$ are related directly by a "tube law" so we don't solve for $p$ and the only quantities that are solved for every timestep are $A$ and $u$.
The tube law has a dissipative term $partial A/partial t$ so through $partial p/partial x$ and the continuity equation, we end up with a $partial^2 u /partial x^2$ term so there's some nice dissipation for the momentum equation.
Now I wanted to quickly solve this using MATLAB's PDEPE and realized that I needed a second spatial derivative for the volume conservation equation as well so I modified it to be
$$fracpartial Apartial t+fracpartialleft(Auright)partial x + epsilonfracpartial^2 Apartial x^2=0$$
where $epsilon$ is really small.
I ran a couple of cases and am seeing nice physically sensible tube deformations and flows. I compute the residual for the original unadulterated volume conservation equation and the values seem acceptable.
But from a theoretical standpoint, I wanted to know how egregious this error is or if it is an acceptable hack for solving the system? I have converted a PDE that had first order spatial derivatives into one that has second order derivatives. Is the modified system completely different or is it the same as the original as $epsilon to 0$? Thanks!
pde numerical-methods systems-of-equations computational-mathematics finite-differences
asked Mar 12 at 21:01
shk92shk92
283
$endgroup$
add a comment |
$begingroup$
I'm trying to solve the system of equations that relates to blood flows in arteries i.e. flow in elastic tubes.
The system looks as follows
$$fracpartial Apartial t+fracpartialleft(Auright)partial x=0$$
$$ fracpartial upartial t+ufracpartial upartial x+frac1rhofracpartial ppartial x=0$$
The first equation is for volume conservation. Fluid velocity is $u$. Pressure $p$ and area $A$ are related directly by a "tube law" so we don't solve for $p$ and the only quantities that are solved for every timestep are $A$ and $u$.
The tube law has a dissipative term $partial A/partial t$ so through $partial p/partial x$ and the continuity equation, we end up with a $partial^2 u /partial x^2$ term so there's some nice dissipation for the momentum equation.
Now I wanted to quickly solve this using MATLAB's PDEPE and realized that I needed a second spatial derivative for the volume conservation equation as well so I modified it to be
$$fracpartial Apartial t+fracpartialleft(Auright)partial x + epsilonfracpartial^2 Apartial x^2=0$$
where $epsilon$ is really small.
I ran a couple of cases and am seeing nice physically sensible tube deformations and flows. I compute the residual for the original unadulterated volume conservation equation and the values seem acceptable.
But from a theoretical standpoint, I wanted to know how egregious this error is or if it is an acceptable hack for solving the system? I have converted a PDE that had first order spatial derivatives into one that has second order derivatives. Is the modified system completely different or is it the same as the original as $epsilon to 0$? Thanks!
pde numerical-methods systems-of-equations computational-mathematics finite-differences
asked Mar 12 at 21:01
shk92shk92
283
$endgroup$
I'm trying to solve the system of equations that relates to blood flows in arteries i.e. flow in elastic tubes.
The system looks as follows
$$fracpartial Apartial t+fracpartialleft(Auright)partial x=0$$
$$ fracpartial upartial t+ufracpartial upartial x+frac1rhofracpartial ppartial x=0$$
The first equation is for volume conservation. Fluid velocity is $u$. Pressure $p$ and area $A$ are related directly by a "tube law" so we don't solve for $p$ and the only quantities that are solved for every timestep are $A$ and $u$.
The tube law has a dissipative term $partial A/partial t$ so through $partial p/partial x$ and the continuity equation, we end up with a $partial^2 u /partial x^2$ term so there's some nice dissipation for the momentum equation.
Now I wanted to quickly solve this using MATLAB's PDEPE and realized that I needed a second spatial derivative for the volume conservation equation as well so I modified it to be
$$fracpartial Apartial t+fracpartialleft(Auright)partial x + epsilonfracpartial^2 Apartial x^2=0$$
where $epsilon$ is really small.
I ran a couple of cases and am seeing nice physically sensible tube deformations and flows. I compute the residual for the original unadulterated volume conservation equation and the values seem acceptable.
But from a theoretical standpoint, I wanted to know how egregious this error is or if it is an acceptable hack for solving the system? I have converted a PDE that had first order spatial derivatives into one that has second order derivatives. Is the modified system completely different or is it the same as the original as $epsilon to 0$? Thanks!
pde numerical-methods systems-of-equations computational-mathematics finite-differences
pde numerical-methods systems-of-equations computational-mathematics finite-differences
asked Mar 12 at 21:01
shk92shk92
283
asked Mar 12 at 21:01
shk92shk92
283
asked Mar 12 at 21:01
shk92shk92
283
asked Mar 12 at 21:01
shk92shk92
283
asked Mar 12 at 21:01
shk92shk92
283
283
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
Usually, addition of an artificial dissipation (AD) term will regularize a numerical solution at the cost of accuracy and it is common practice in computational methods. In some cases, it is even necessary to employ AD to obtain a solution for a problem, but this may depend on the discretization scheme you are using.
For example, at transonic flows with shock waves a central difference FV scheme that employs higher order scalar dissipation at smooth regions and 1st order scalar dissipation near shock-waves is very popular in aeronautics (Jameson-Schmid-Turkel scheme) and important to avoid the odd-even decoupling of the central scheme. Other, more advanced options are to use anisotropic scaling of the dissipation coefficient with respect to local flow properties. For instance, matrix dissipation schemes are used in flows with boundary layers where the AD terms are scaled appropriately to obtain sharper boundary layer gradients.
So, informally speaking, you need $epsilon$ small enough for accuracy but large enough for stability and this $epsilon$ may strongly depend on local flow gradients/Jacobians and is problem depended.
What you are asking I think is a bit general, but there is the theory of viscosity solutions and the method of vanishing viscosity which is the study of how some estimates of your modified (approximate) solution $(u^epsilon, A^epsilon)$ are affected as $epsilon to 0$. You may want to check Evan's PDE book Chapter 10 and its references.
answered Mar 13 at 0:56
aresares
202110
$endgroup$
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
add a comment |
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1 Answer
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active
oldest
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1 Answer
1
active
oldest
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active
oldest
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active
oldest
votes
$begingroup$
Usually, addition of an artificial dissipation (AD) term will regularize a numerical solution at the cost of accuracy and it is common practice in computational methods. In some cases, it is even necessary to employ AD to obtain a solution for a problem, but this may depend on the discretization scheme you are using.
For example, at transonic flows with shock waves a central difference FV scheme that employs higher order scalar dissipation at smooth regions and 1st order scalar dissipation near shock-waves is very popular in aeronautics (Jameson-Schmid-Turkel scheme) and important to avoid the odd-even decoupling of the central scheme. Other, more advanced options are to use anisotropic scaling of the dissipation coefficient with respect to local flow properties. For instance, matrix dissipation schemes are used in flows with boundary layers where the AD terms are scaled appropriately to obtain sharper boundary layer gradients.
So, informally speaking, you need $epsilon$ small enough for accuracy but large enough for stability and this $epsilon$ may strongly depend on local flow gradients/Jacobians and is problem depended.
What you are asking I think is a bit general, but there is the theory of viscosity solutions and the method of vanishing viscosity which is the study of how some estimates of your modified (approximate) solution $(u^epsilon, A^epsilon)$ are affected as $epsilon to 0$. You may want to check Evan's PDE book Chapter 10 and its references.
answered Mar 13 at 0:56
aresares
202110
$endgroup$
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
add a comment |
$begingroup$
Usually, addition of an artificial dissipation (AD) term will regularize a numerical solution at the cost of accuracy and it is common practice in computational methods. In some cases, it is even necessary to employ AD to obtain a solution for a problem, but this may depend on the discretization scheme you are using.
For example, at transonic flows with shock waves a central difference FV scheme that employs higher order scalar dissipation at smooth regions and 1st order scalar dissipation near shock-waves is very popular in aeronautics (Jameson-Schmid-Turkel scheme) and important to avoid the odd-even decoupling of the central scheme. Other, more advanced options are to use anisotropic scaling of the dissipation coefficient with respect to local flow properties. For instance, matrix dissipation schemes are used in flows with boundary layers where the AD terms are scaled appropriately to obtain sharper boundary layer gradients.
So, informally speaking, you need $epsilon$ small enough for accuracy but large enough for stability and this $epsilon$ may strongly depend on local flow gradients/Jacobians and is problem depended.
What you are asking I think is a bit general, but there is the theory of viscosity solutions and the method of vanishing viscosity which is the study of how some estimates of your modified (approximate) solution $(u^epsilon, A^epsilon)$ are affected as $epsilon to 0$. You may want to check Evan's PDE book Chapter 10 and its references.
answered Mar 13 at 0:56
aresares
202110
$endgroup$
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
add a comment |
$begingroup$
Usually, addition of an artificial dissipation (AD) term will regularize a numerical solution at the cost of accuracy and it is common practice in computational methods. In some cases, it is even necessary to employ AD to obtain a solution for a problem, but this may depend on the discretization scheme you are using.
For example, at transonic flows with shock waves a central difference FV scheme that employs higher order scalar dissipation at smooth regions and 1st order scalar dissipation near shock-waves is very popular in aeronautics (Jameson-Schmid-Turkel scheme) and important to avoid the odd-even decoupling of the central scheme. Other, more advanced options are to use anisotropic scaling of the dissipation coefficient with respect to local flow properties. For instance, matrix dissipation schemes are used in flows with boundary layers where the AD terms are scaled appropriately to obtain sharper boundary layer gradients.
So, informally speaking, you need $epsilon$ small enough for accuracy but large enough for stability and this $epsilon$ may strongly depend on local flow gradients/Jacobians and is problem depended.
What you are asking I think is a bit general, but there is the theory of viscosity solutions and the method of vanishing viscosity which is the study of how some estimates of your modified (approximate) solution $(u^epsilon, A^epsilon)$ are affected as $epsilon to 0$. You may want to check Evan's PDE book Chapter 10 and its references.
answered Mar 13 at 0:56
aresares
202110
$endgroup$
Usually, addition of an artificial dissipation (AD) term will regularize a numerical solution at the cost of accuracy and it is common practice in computational methods. In some cases, it is even necessary to employ AD to obtain a solution for a problem, but this may depend on the discretization scheme you are using.
For example, at transonic flows with shock waves a central difference FV scheme that employs higher order scalar dissipation at smooth regions and 1st order scalar dissipation near shock-waves is very popular in aeronautics (Jameson-Schmid-Turkel scheme) and important to avoid the odd-even decoupling of the central scheme. Other, more advanced options are to use anisotropic scaling of the dissipation coefficient with respect to local flow properties. For instance, matrix dissipation schemes are used in flows with boundary layers where the AD terms are scaled appropriately to obtain sharper boundary layer gradients.
So, informally speaking, you need $epsilon$ small enough for accuracy but large enough for stability and this $epsilon$ may strongly depend on local flow gradients/Jacobians and is problem depended.
What you are asking I think is a bit general, but there is the theory of viscosity solutions and the method of vanishing viscosity which is the study of how some estimates of your modified (approximate) solution $(u^epsilon, A^epsilon)$ are affected as $epsilon to 0$. You may want to check Evan's PDE book Chapter 10 and its references.
answered Mar 13 at 0:56
aresares
202110
answered Mar 13 at 0:56
aresares
202110
answered Mar 13 at 0:56
aresares
202110
answered Mar 13 at 0:56
aresares
202110
202110
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
add a comment |
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
Thank you so much for your answer! I've just obtained the Lawrence Evans book that has a lot of useful stuff for my problem like you said. I'll read it over the weekend. So essentially, the addition of my term is not causing any fundamental change in the physics right? That's the main thing I'm worried about... Accuracy is not my concern right now.
$endgroup$
– shk92
Mar 13 at 4:02
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
$begingroup$
You're welcome. My answer, however, was of general nature. I could only speculate that for small enough $epsilon$ there will be no problem, provided that your "tube law" is smooth and the same holds for the solution. If you cannot perform some kind of analytic study, then I would suggest you to test different discretization methods that they will not require the addition of an AD term and compare.
$endgroup$
– ares
Mar 13 at 4:31
add a comment |
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