8 votes

why not all conservation laws solved numerically by hyperbolic methods

The core difference between the two PDEs you presented is that the heat equation is parabolic while Burgers equation is hyperbolic. This means that for Burgers equation changes in the solution travel ...
Dan Doe's user avatar
  • 1,083
6 votes

why not all conservation laws solved numerically by hyperbolic methods

Actually, the heat equation can be, and often is, solved by hyperbolic methods. Instead of writing $q=-u_x$, write $q_t=-(u_x+q)/\tau$. Instead of the heat flux becoming instantaneously equal to the ...
Philip Roe's user avatar
  • 1,154
5 votes
Accepted

How to implement Lax-Friedrich flux splitting with WENO scheme

I think you mean with Lax-Friedrichs the local Lax-Friedrichs or Rusanov scheme [1], [2] where the wave speeds (left & right-going) are given by the maximum eigenvalue $\lambda_\max := \max_i | \...
Dan Doe's user avatar
  • 1,083
4 votes

Physical interpretation of divergence theorem

The condition that has to be satisfied at the node point 2 is that the influx equals the outflux. Here, this will then be $$ v_{12}C_{12} = v_{24}C_{24}+v_{23}C_{23}, $$ but $C_{ij}$ is not the ...
Wolfgang Bangerth's user avatar
3 votes
Accepted

Why is the maximum potential energy greater than the maximum kinetic energy?

You've defined your potential energy incorrectly. Your spring forcing function is $$ F = b x + w^2 x^{p-1} $$ The change in potential energy is defined as $$ \Delta V = \int_{x_0}^{x_1} F dx = \left.\...
helloworld922's user avatar
3 votes

Non-conservative advective term in a finite volume scheme

There is a methology for non-conservative products called path-conservative schemes, which might useful for you. The method can be applied to systems of the form \begin{align} \frac{\partial \mathbf{Q}...
ConvexHull's user avatar
  • 1,286
3 votes
Accepted

Question about energy in the shallow water equations on a staggered grid

I'm going to work on the assumption that the disturbances aren't large enough to create shock waves, which are a whole other can of worms. Your energy functional is close but not quite right -- the ...
Daniel Shapero's user avatar
3 votes
Accepted

Numerical methods that can be written in flux conservative form

The key feature to a conservative method is simply that the changes due to the fluxes cancel out (i.e., the flux leaving one cell is entering another), so the total mass is constant. Using the form ...
David Ketcheson's user avatar
3 votes

Conservation violation in axisymmetric Diffusion Equation

You may want a method that works right down to the coordinate singularity at $r=0$. I will do the spherical case, but the cylindrical case is similar. We want to avoid ever dividing by $r$. To think ...
Philip Roe's user avatar
  • 1,154
2 votes
Accepted

Discrete conservation and Finite Element methods

Most pdes coming from physics have a divergence structure $$ u_t + \nabla\cdot F =0 \qquad \textrm{in} \quad \Omega $$ Then for any arbitrary control volume $D \subset \Omega$, we have $$ \frac{d}{...
cfdlab's user avatar
  • 3,028
2 votes

Conservatives in shock tube

Will it give discontinuous result in flux terms? Yes, the flux terms in general may be discontinuous. Is it because shock tube problem is transient and equation $\rho_1 u_1= \rho_2 u_2$ is steady? ...
helloworld922's user avatar
2 votes
Accepted

Numerical Lax-Wendroff scheme order of convergence on Burgers equation

Probably not. If you have inviscid Burgers equation then your discontinuous initial condition should (somewhere) stay discontinuous because there is no viscosity. If there is discontinuity in the ...
lmalenica's user avatar
2 votes
Accepted

Mass conservation for hyperbolic relaxation problem

Yes, mass conservation holds, because in fact you are solving some sort of a relativistic diffusion equation. Why: $$\partial_{t} u + \partial_{x} v = 0$$ and: $$\partial_{t} v + \frac{1}{\epsilon^{...
Mithridates the Great's user avatar
2 votes

Conservative interpolation from a 1D grid to another 1D grid

I'll add some thoughts and terminology. First, your question doesn't make that much sense or is a bit underspecified. You are given a function values on a grid $\{x_i\}$, and you want the ...
davidhigh's user avatar
  • 3,127
1 vote

Conservative interpolation from a 1D grid to another 1D grid

You can't do this with interpolation. Imagine for example that you are representing the function $f(x)=x^2$ on the interval $[-1,1]$ on a mesh $X_{old}$ with a very small mesh width. This ...
Wolfgang Bangerth's user avatar
1 vote

Mass conservation for hyperbolic relaxation problem

It's clear that the mass of $v$ is not conserved in general. Just take $u(x,t=0) = 0$ and choose $f$ so that $f(0)=0$. Then $$\left. \frac{\partial v}{\partial t}\right|_{t=0} = -\frac{v}{\epsilon^...
David Ketcheson's user avatar
1 vote

Should energy be conserved in an N-body simulation where particles don't lose energy in collisions?

The numerical integration scheme might not be energy-preserving. The error term is often strictly positive, allowing for non-secular growth of the energy. There are some ways around this, such as ...
Malcolm's user avatar
  • 169
1 vote
Accepted

Should energy be conserved in an N-body simulation where particles don't lose energy in collisions?

Think of the collisions between particles or walls as being modeled by a potential energy term of the form: $$U(r) = \begin{cases} 0, & \text{if $r > r_c$}, \\ +\infty, & \text{if $r \...
Juan M. Bello-Rivas's user avatar

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