Solving second order time-dependent PDE
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 '18 at 17:46
Harry49
5,99121031
asked Nov 21 '18 at 16:39
math123
163
|
show 2 more comments
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 '18 at 17:46
Harry49
5,99121031
asked Nov 21 '18 at 16:39
math123
163
Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 at 17:21
|
show 2 more comments
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 '18 at 17:46
Harry49
5,99121031
asked Nov 21 '18 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 '18 at 17:46
Harry49
5,99121031
asked Nov 21 '18 at 16:39
math123
163
edited Nov 23 '18 at 17:46
Harry49
5,99121031
asked Nov 21 '18 at 16:39
math123
163
edited Nov 23 '18 at 17:46
Harry49
5,99121031
edited Nov 23 '18 at 17:46
Harry49
5,99121031
edited Nov 23 '18 at 17:46
Harry49
5,99121031
5,99121031
asked Nov 21 '18 at 16:39
math123
163
asked Nov 21 '18 at 16:39
math123
163
asked Nov 21 '18 at 16:39
math123
163
163
Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 at 17:21
|
show 2 more comments
Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 at 17:21
Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 at 16:45
Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 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 '18 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 '18 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 '18 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 '18 at 17:21
|
show 2 more comments
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Why not just method of lines in direction $t$?
– Federico
Nov 21 '18 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 '18 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 '18 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 '18 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 '18 at 17:21