All Questions
Tagged with finite-difference hyperbolic-pde
46
questions
3
votes
1
answer
270
views
Why does the FDM give a correct solution to a PDE with a discontinuous initial condition?
I was solving the dimensionless wave equation:
$$ u_{xx} = u_{tt} \tag 1$$
with the initial conditions:
$$ u(x,0) = 0 \tag 2 $$
$$ u_t(x=0,0) = v_0 \tag 2 $$
$$ u_t(x>0,0) = 0 \tag 3 $$
and the ...
2
votes
1
answer
132
views
Can the Runge-Kuta algorithm help in reducing numerical dispersion and anisotropy when using the FDM to solve the 2D wave equation? [closed]
I am currently studying the effects of group velocity on the finite difference solution of the wave equation. Most of what I learned is from this source. I understand that high frequency components in ...
- 45
1
vote
0
answers
102
views
Can I reduce my simulation error with a staggered grid, postprocessing and compatibility equation feedback?
What I did
Using the finite difference method, I solved with a certain amount of error the following system of hyperbolic partial differential equations in cylindrical coordinates (the problem is ...
2
votes
0
answers
102
views
Numerical solution for inviscid Burgers' equation seems to have no breaking time?
So I'm trying to use the Lax-Friedrichs method to solve the inviscid burgers' equation with initial condition $$u(x,0) = \sin(x)$$, using
$$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
- 129
0
votes
0
answers
45
views
Which numerical method can I use to solve this system of hyperbolic PDEs?
Backround
The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
1
vote
1
answer
102
views
Simulating First Order Hyperbolic PDE with Finite Difference Scheme
I am trying to simulate a hyperbolic PDE with some control given by the following:
$$u_t(x, t) = u_x(x, t) + \theta(x) u(0, t)$$
with boundary conditions:
$$u(1, t) = U(t) = \int_0^1 k(1-y) u(y, t)dy$$...
- 31
0
votes
0
answers
83
views
Solving a system of PDEs with an ODE
I want to solve the following system of equations which consists two PDEs and one ODE:
\begin{align}
\rho_t+v\rho_x &= 0; \newline
Y_t+vY_x &= 0 ;\newline
v_t &= -\frac{1}{(\...
- 11
5
votes
1
answer
245
views
Solving simplified 1D plasma fluid equations with finite difference
The following two equations represent a simple model of a plasma where ions are immobile.
$$n\frac{\partial u}{\partial t}+nu\frac{\partial u}{\partial x}=n\frac{d\phi}{dx}-\theta\frac{\partial n}{\...
- 151
6
votes
1
answer
301
views
How can I check mass conservation when solving the advection equation using an upwind scheme?
My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background
Consider ...
- 317
2
votes
1
answer
579
views
Numerical solution of 2D wave equation using Fourier transform and finite differences
This is the $2$-dimensional wave equation
$$ u_{tt} = u_{xx} + u_{yy} $$
with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$.
The inverse Fourier transform used is
$$ u(x,y,t) = \iint \hat{...
- 121
2
votes
1
answer
211
views
Shock Capturing Methods for Shallow Water Equations
I am looking for some help finding a numerical solution to the shallow water equations:
$\partial_tu+\partial_x(u^2/2+g\eta)=0$
$\partial_t \eta+\partial_x(u\eta)=0$.
where $u$ is the depth averaged ...
- 23
0
votes
0
answers
51
views
How do I identify negative group speeds?
This question is a continuation of one of my other questions.
I've been trying to show that collocated (non-staggered) grids can suffer from negative group speeds in the linearized shallow water ...
- 105
0
votes
1
answer
166
views
A simple wave for the linear shallow water equations
I'm looking for a simple, right-traveling wave for the linear shallow water equations (1D). My question: what are the initial conditions (velocity $U_0(x)$ and/or average water height, average ...
- 105
3
votes
0
answers
54
views
Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?
I am trying to simulate the following equation using FDTD
$ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $
subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
- 131
3
votes
0
answers
214
views
TVD Lax-Wendroff with non-constant velocity
I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D.
The equation is the following:
\begin{equation}
\frac{\...
- 319