# Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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### Why is local conservation important when solving PDEs?

Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs. What can go wrong when using a ...
16k views

### Recommendation for Finite Difference Method in Scientific Python

For a project I am working on (in hyperbolic PDEs) I would like to get some rough handle on the behavior by looking at some numerics. I am, however, not a very good programmer. Can you recommend ...
1k views

### When should implicit methods be used in the integration of hyperbolic PDEs?

Numerical methods for solving PDEs (or ODEs) fall into two broad categories: explicit and implicit methods. Implicit methods allow larger stable timesteps but require more work per step. For ...
8k views

### Understanding the Courant–Friedrichs–Lewy condition

I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration ...
857 views

### How can I choose a good Riemann solver when numerically solving a system of hyperbolic PDEs?

Many numerical methods for hyperbolic PDEs are based on the use of Riemann solvers. Such solvers are essential for accurately capturing shock waves. There are a range of such solvers available for ...
371 views

### How to construct well-balanced finite volume and discontinuous Galerkin methods for hyperbolic PDEs with source terms?

Source terms, such as those due to bathymetry in the shallow water equations, need to be integrated in a special way in order to preserve physical steady states. Is there a general way to construct ...
657 views

### Conservative finite-difference expression for the advection equation

Following on from the earlier question I am trying to derive a finite-difference scheme for the advection equation which is conservative. It was suggested that for advection equation with variable ...
269 views

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

I am trying to think of what this kind of problem is called. I illustrate it with a telegrapher's equation with (hopefully) standard notation. Find $u:\Omega\times \mathbb{R} \to \mathbb{R}$ such ...
367 views

### Euler Equation Eigensystem with Gravity in the Energy Flux

I am modifying a conservative form of the Euler equations with gravity in the energy flux (see previous question: Energy Conservation in Conservation Laws with Source Terms) for use in a Riemann ...
241 views

### FVM - virtual node discretisation

I have come across the paper titled: A monolithic fluid structure interaction algorithm ... For a 1D grid, at the boundaries the paper uses virtual nodes $x_{0}$ and $x_{N+1}$ (page 372) and for ...
422 views

### Energy Conservation in Conservation Laws with Source Terms

I'm wondering if anyone can help me understand energy conservation when using conservation law methods (i.e. Riemann solver, High-Resolution Wave-Propagation Methods) with the addition of source terms:...
I am implementing the advection equation $u_x+(1/c)u_t=0$ following a Crank-Nicholson finite difference scheme. The equation for this is \begin{eqnarray*} -\frac{\gamma}{4} w_{n-3 j+1} + w_{n-2 j+1} ...
Suppose we have the inhomogeneous advection equation $$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$ for $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...