Solving PDEs by adding small perturbation terms and taking limits
When trying to solve some PDE problem, sometimes we made approximation problem that is easier to solve than the original one. We do that by adding a new term(s) in the original problem. Then we try to use the solution of the approximation problem to get the solution of the original problem.
Well known method for doing something like this is the vanishing viscosity method. The vanishing viscosity method consists in viewing problem
$$(OP) hspace{1cm}
u_t+g(u)_x = 0\[2ex]
$$
as the limit of the problem
$$(AP) hspace{1cm}
u_t+g(u)_x =nu varDelta u \[2ex]
$$
when $nu rightarrow 0 $. So we could find solution of a original problem by finding the solution of the approximation problem by letting $nu rightarrow 0 $. Additional info on the vanishing viscosity method could be found in a lot of books (Dafermos, for example).
Before asking the question, just to mention that my areas of interest are conservation and balance laws, and I work more with systems than with single equations.
My question is this: what are the other possible choices we can use other than $nu varDelta u$ for the term we add in the original problem?
I am especially interested in the cases where we let $nu rightarrow 0$ in the approximation problem and then make some conclusion about solution of the original problem.
I found a few papers that deal with PDE problems with relaxation (for example, in case of 2x2 systems, they add perturbation terms $(phi (u_2)-u_1)/nu$ and then try to find zero relaxation limit by letting $nu rightarrow 0 $ ) and PDE problems with damping (they add some terms like $nu cdot u$ but they never let $nu rightarrow 0 $). Also problems with damping are more interesting to me than with the relaxation.
Does anyone know any reference in the literature with some other choices other the few mentioned above? Also it would be very helpful to me if someone knows some work that deals with PDE problem with damping and in some moment authors let $nu rightarrow 0 $.
reference-request pde relaxations
asked Dec 9 '18 at 21:28
Mark
4617
add a comment |
When trying to solve some PDE problem, sometimes we made approximation problem that is easier to solve than the original one. We do that by adding a new term(s) in the original problem. Then we try to use the solution of the approximation problem to get the solution of the original problem.
Well known method for doing something like this is the vanishing viscosity method. The vanishing viscosity method consists in viewing problem
$$(OP) hspace{1cm}
u_t+g(u)_x = 0\[2ex]
$$
as the limit of the problem
$$(AP) hspace{1cm}
u_t+g(u)_x =nu varDelta u \[2ex]
$$
when $nu rightarrow 0 $. So we could find solution of a original problem by finding the solution of the approximation problem by letting $nu rightarrow 0 $. Additional info on the vanishing viscosity method could be found in a lot of books (Dafermos, for example).
Before asking the question, just to mention that my areas of interest are conservation and balance laws, and I work more with systems than with single equations.
My question is this: what are the other possible choices we can use other than $nu varDelta u$ for the term we add in the original problem?
I am especially interested in the cases where we let $nu rightarrow 0$ in the approximation problem and then make some conclusion about solution of the original problem.
I found a few papers that deal with PDE problems with relaxation (for example, in case of 2x2 systems, they add perturbation terms $(phi (u_2)-u_1)/nu$ and then try to find zero relaxation limit by letting $nu rightarrow 0 $ ) and PDE problems with damping (they add some terms like $nu cdot u$ but they never let $nu rightarrow 0 $). Also problems with damping are more interesting to me than with the relaxation.
Does anyone know any reference in the literature with some other choices other the few mentioned above? Also it would be very helpful to me if someone knows some work that deals with PDE problem with damping and in some moment authors let $nu rightarrow 0 $.
reference-request pde relaxations
asked Dec 9 '18 at 21:28
Mark
4617
add a comment |
When trying to solve some PDE problem, sometimes we made approximation problem that is easier to solve than the original one. We do that by adding a new term(s) in the original problem. Then we try to use the solution of the approximation problem to get the solution of the original problem.
Well known method for doing something like this is the vanishing viscosity method. The vanishing viscosity method consists in viewing problem
$$(OP) hspace{1cm}
u_t+g(u)_x = 0\[2ex]
$$
as the limit of the problem
$$(AP) hspace{1cm}
u_t+g(u)_x =nu varDelta u \[2ex]
$$
when $nu rightarrow 0 $. So we could find solution of a original problem by finding the solution of the approximation problem by letting $nu rightarrow 0 $. Additional info on the vanishing viscosity method could be found in a lot of books (Dafermos, for example).
Before asking the question, just to mention that my areas of interest are conservation and balance laws, and I work more with systems than with single equations.
My question is this: what are the other possible choices we can use other than $nu varDelta u$ for the term we add in the original problem?
I am especially interested in the cases where we let $nu rightarrow 0$ in the approximation problem and then make some conclusion about solution of the original problem.
I found a few papers that deal with PDE problems with relaxation (for example, in case of 2x2 systems, they add perturbation terms $(phi (u_2)-u_1)/nu$ and then try to find zero relaxation limit by letting $nu rightarrow 0 $ ) and PDE problems with damping (they add some terms like $nu cdot u$ but they never let $nu rightarrow 0 $). Also problems with damping are more interesting to me than with the relaxation.
Does anyone know any reference in the literature with some other choices other the few mentioned above? Also it would be very helpful to me if someone knows some work that deals with PDE problem with damping and in some moment authors let $nu rightarrow 0 $.
reference-request pde relaxations
asked Dec 9 '18 at 21:28
Mark
4617
When trying to solve some PDE problem, sometimes we made approximation problem that is easier to solve than the original one. We do that by adding a new term(s) in the original problem. Then we try to use the solution of the approximation problem to get the solution of the original problem.
Well known method for doing something like this is the vanishing viscosity method. The vanishing viscosity method consists in viewing problem
$$(OP) hspace{1cm}
u_t+g(u)_x = 0\[2ex]
$$
as the limit of the problem
$$(AP) hspace{1cm}
u_t+g(u)_x =nu varDelta u \[2ex]
$$
when $nu rightarrow 0 $. So we could find solution of a original problem by finding the solution of the approximation problem by letting $nu rightarrow 0 $. Additional info on the vanishing viscosity method could be found in a lot of books (Dafermos, for example).
Before asking the question, just to mention that my areas of interest are conservation and balance laws, and I work more with systems than with single equations.
My question is this: what are the other possible choices we can use other than $nu varDelta u$ for the term we add in the original problem?
I am especially interested in the cases where we let $nu rightarrow 0$ in the approximation problem and then make some conclusion about solution of the original problem.
I found a few papers that deal with PDE problems with relaxation (for example, in case of 2x2 systems, they add perturbation terms $(phi (u_2)-u_1)/nu$ and then try to find zero relaxation limit by letting $nu rightarrow 0 $ ) and PDE problems with damping (they add some terms like $nu cdot u$ but they never let $nu rightarrow 0 $). Also problems with damping are more interesting to me than with the relaxation.
Does anyone know any reference in the literature with some other choices other the few mentioned above? Also it would be very helpful to me if someone knows some work that deals with PDE problem with damping and in some moment authors let $nu rightarrow 0 $.
reference-request pde relaxations
reference-request pde relaxations
asked Dec 9 '18 at 21:28
Mark
4617
asked Dec 9 '18 at 21:28
Mark
4617
asked Dec 9 '18 at 21:28
Mark
4617
asked Dec 9 '18 at 21:28
Mark
4617
asked Dec 9 '18 at 21:28
Mark
4617
4617
add a comment |
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