Solving second order time-dependent PDE
up vote
1
down vote
favorite
I am trying to solve a non-linear second order PDE as given below.
$$partial_{t}u+upartial_{x}u=partial_{xx}u+u$$
with periodic boundary condition: $$u(x,t)=u(x+2pi,t)$$ and initial condition: $$u(x,0)=u_{0}(x)$$
I am using a fractional step method to solve this problem as shown below.
- 1st step: Non-linear part $$partial_{t}u=-upartial_{x}u$$
- 2nd step: Linear part $$partial_{t}u=partial_{xx}u+u$$
using Fourier discretization in space.
But I do not get reasonable results and there are oscillations.
Any suggestions for any other method for solving such PDEs?
pde numerical-methods
edited Nov 23 at 17:46
Harry49
5,6302929
asked Nov 21 at 16:39
math123
163
|
show 2 more comments
up vote
1
down vote
favorite
I am trying to solve a non-linear second order PDE as given below.
$$partial_{t}u+upartial_{x}u=partial_{xx}u+u$$
with periodic boundary condition: $$u(x,t)=u(x+2pi,t)$$ and initial condition: $$u(x,0)=u_{0}(x)$$
I am using a fractional step method to solve this problem as shown below.
- 1st step: Non-linear part $$partial_{t}u=-upartial_{x}u$$
- 2nd step: Linear part $$partial_{t}u=partial_{xx}u+u$$
using Fourier discretization in space.
But I do not get reasonable results and there are oscillations.
Any suggestions for any other method for solving such PDEs?
pde numerical-methods
edited Nov 23 at 17:46
Harry49
5,6302929
asked Nov 21 at 16:39
math123
163
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21
|
show 2 more comments
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I am trying to solve a non-linear second order PDE as given below.
$$partial_{t}u+upartial_{x}u=partial_{xx}u+u$$
with periodic boundary condition: $$u(x,t)=u(x+2pi,t)$$ and initial condition: $$u(x,0)=u_{0}(x)$$
I am using a fractional step method to solve this problem as shown below.
- 1st step: Non-linear part $$partial_{t}u=-upartial_{x}u$$
- 2nd step: Linear part $$partial_{t}u=partial_{xx}u+u$$
using Fourier discretization in space.
But I do not get reasonable results and there are oscillations.
Any suggestions for any other method for solving such PDEs?
pde numerical-methods
edited Nov 23 at 17:46
Harry49
5,6302929
asked Nov 21 at 16:39
math123
163
I am trying to solve a non-linear second order PDE as given below.
$$partial_{t}u+upartial_{x}u=partial_{xx}u+u$$
with periodic boundary condition: $$u(x,t)=u(x+2pi,t)$$ and initial condition: $$u(x,0)=u_{0}(x)$$
I am using a fractional step method to solve this problem as shown below.
- 1st step: Non-linear part $$partial_{t}u=-upartial_{x}u$$
- 2nd step: Linear part $$partial_{t}u=partial_{xx}u+u$$
using Fourier discretization in space.
But I do not get reasonable results and there are oscillations.
Any suggestions for any other method for solving such PDEs?
pde numerical-methods
pde numerical-methods
edited Nov 23 at 17:46
Harry49
5,6302929
asked Nov 21 at 16:39
math123
163
edited Nov 23 at 17:46
Harry49
5,6302929
asked Nov 21 at 16:39
math123
163
edited Nov 23 at 17:46
Harry49
5,6302929
edited Nov 23 at 17:46
Harry49
5,6302929
edited Nov 23 at 17:46
Harry49
5,6302929
5,6302929
asked Nov 21 at 16:39
math123
163
asked Nov 21 at 16:39
math123
163
asked Nov 21 at 16:39
math123
163
163
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21
|
show 2 more comments
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21
|
show 2 more comments
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3007977%2fsolving-second-order-time-dependent-pde%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Why not just method of lines in direction $t$?
– Federico
Nov 21 at 16:45
Also, the Cauchy problem for this PDE doesn't seem always very well posed. I suspect you get only local existence in time.
– Federico
Nov 21 at 16:53
Consider for instance the simpler $partial_t u = u(1-partial_t u)$. If $partial_tu>-1$, then solutions try to grow exponentially and then I don't know if your PDE blows up...
– Federico
Nov 21 at 16:55
If your PDE is not well behaved, no matter how good a numerical scheme you pick, it will never avoid blow-ups
– Federico
Nov 21 at 16:56
You are asking if there is one of the members who can solve the $1$-D Navier-Stokes equation...
– Daniele Tampieri
Nov 21 at 17:21