12 votes
Accepted

Why do we solve non-linearity in hyperbolic PDEs that way?

The good thing about the conservative form is that this comprises multiple models, such as Shallow Water Equations, Euler Equations or traffic models. An essential feature of hyperbolic equations is ...
  • 671
8 votes
Accepted

Understanding the Courant–Friedrichs–Lewy condition

I have two extra points I would like to add to Wolfgang's answer. A formulation of the CFL condition that I find more useful than the classic formula is this: A necessary condition for the ...
7 votes
Accepted

Do the class of PDEs that lack initial conditions have a name?

Such problems (sometimes called lateral Cauchy problems) are in general not well-posed (meaning they either lack a solution, or there are infinitely many of them, or the solution is unstable under ...
7 votes

Understanding the Courant–Friedrichs–Lewy condition

You are correct: If you satisfy the CFL condition, then all that guarantees is that your scheme is stable, i.e., the numerical solution does not go to infinity. But the CFL condition says nothing ...
6 votes
Accepted

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

Let me give a part of your answer, I would need some more indications from your side to answer you fully. So please read, and write some comments so that I can complete my answer. About notations in ...
6 votes
Accepted

When is it safe to ignore the diffusion term in an advection-diffusion equation?

The stationary equation you show transports information from the right to the left via the advection term; it also diffuses slightly. If you switch off the diffusion term altogether, then you only ...
5 votes
Accepted

Hyperbolic Equation PDE (Python)

First-order hyperbolic equations model conservation laws; as the alternative name "transport equations" suggests, they transport information along so-called "characteristic curves" with a finite speed ...
5 votes

CFL condition for variable coefficients

The CFL number (or simply Courant number) is defined locally: $\nu = u(x,t) \frac{\Delta t}{\Delta x}$. The velocity may vary in time and space and the grid may be non-uniform (in $x$ and/or $t$). ...
5 votes

what do zero real parts of eigenvalues mean? Any good references?

Eigenvalues with zero eigenvalue correspond to purely oscillatory modes. You can see it by diagonalising the system. Your matrix $A$ can be written as \begin{equation} A = P \Sigma P^{-1} \end{...
  • 1,208
5 votes
Accepted

How to set up a shock tube problem such that the solution includes a shock with a specified Mach number

I have implemented the solution derived below using PyClaw in an IPython notebook. If you download that, you can adjust the initial values and see the computed solutions. General setup In the ...
5 votes
Accepted

Crank-Nicolson method for inhomogeneous advection equation

Your equation can be written in the following fashion (any spatial derivative approximation is valid), once space is discretised: $$\frac{1}{c}\frac{du_i}{dt}=-\left(\frac{\partial u}{\partial x}\...
  • 1,606
5 votes
Accepted

Is this the correct way for solving coupled 1d PDEs using finite difference methods?

This doesn't answer your question directly but instead suggests an alternate strategy. In general, unless you are interested in experimenting with numerical methods, I recommend using an existing PDE ...
  • 5,784
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 | \...
  • 671
5 votes

stability of a numercial scheme for a hyperbolic system?

It is worth making some additional points. What you set out is just one version of the Lax-Wendroff method. That scheme is unique in one space dimension but has several free parameters in two or three ...
4 votes

What are the good testing problems for hyperbolic equation?

The first two can be decomposed into a system of first-order advection equations of which there are a number of standard test problems often involving both smooth and discontinuous initial conditions ...
4 votes
Accepted

Why have specialised upwind schemes been developed to solve hyperbolic equations?

To elaborate on Wolfgang's answer: since hyperbolic PDE semi-discretizations with centered differences have purely imaginary eigenvalues, they are only neutrally stable. For linear problems (e.g., ...
4 votes

Why have specialised upwind schemes been developed to solve hyperbolic equations?

Central differencing schemes are not stable if you have advection dominated problems. There really was no other trivial [1] alternative to developing upwind schemes. [1] There are a few other ...
4 votes

How to derive the stability of time stepping schemes?

Another way you can find stability is to assume first you are solving the following model problem for complex $\lambda$: $$\dot{x} = f(t,x) = \lambda x$$ with exact solution: $x(t) = x_0 e^{\lambda t}$...
  • 3,733
4 votes
Accepted

Numerical methods for the $u_t + \frac{(u_x)^2}{2} = 0$ equation

This is a Hamilton-Jacobi equation. You can read about how to apply WENO to such equations in Section 4 of Chi-Wang Shu's 2009 WENO review paper, and references therein.
4 votes
Accepted

Grid dependence of a numerical model

Your numerical solution is probably just getting more accurate as you increase the number of grid points. Do you know or have you tried to derive the analytic (exact) solution for this problem? By ...
  • 323
4 votes
Accepted

Shallow water equations (SWE): well-posed initial data for single travelling pulse

What you are looking for is known as a simple wave solution, in which the variation in the solution belongs entirely to one characteristic family. If you had a linear hyperbolic system, this would ...
4 votes

Problems with manufactured solutions for 1D inviscid burgers' equation

You simply have a bug in your code. The flux is $\frac{1}{2} u^2$ and not $\frac{1}{4} u^2$.
3 votes

How to derive the stability of time stepping schemes?

One way is via von Neumann stability analysis. This technique hinges on the fact that the Fourier transform can sometimes be regarded as an $L^2$ isometry. This and other techniques such as direct ...
3 votes
Accepted

upwind schemes for solving inviscid euler equations

Do I definitely need to know the eigenvectors/eigenvalues of the system if I need to use an upwind scheme for such flow? No, it is certainly not necessary to use the full eigenstructure of the system,...
3 votes

what do zero real parts of eigenvalues mean? Any good references?

In the 2-D case, this corresponds to an elliptic fixed point (an orbital, I believe). You might look into Lyapunov stability, hopefully someone will be able to recommend a good resource on that. I'...
3 votes
Accepted

Energy Conservation in Conservation Laws with Source Terms

I have not studied this particular system before, and I'm sure that someone who has could say much more than I will. I don't think there is any reason to expect that discretizing the equations in the ...
3 votes

Why is local conservation important when solving PDEs?

Many times, the equations to be solved represent a physical conservation law. For example, the Euler equations for fluid dynamics are representations of conservation of mass, momentum, and energy. ...
  • 163
3 votes

How can I solve this 1D nonlinear, variable-coefficient hyperbolic PDE?

This is just a small addition to David Ketcheson's comments about solving numerically. If you want to obtain a numerical solution using MATLAB, Lawrence Shampine has written a MATLAB solver for ...
  • 5,784
3 votes

Wave Equation PDE

The problem is that odeint solves first-order ODEs of the form $$\dot u(t) = Au(t),$$ but the wave equation is (after discretization in space) a second-order ODE of ...
3 votes

Numerical quadrature in Discontinuous Galerkin

Which basis you use for your finite element space does not matter for quadrature in general. If you use, for example, polynomials of degree $k$ (whether the Lagrange basis, or the Legendre basis, or ...

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