# 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 ...
616 views

### Which methods can ensure that physical quantities remain positive throughout a PDE simulation?

Physical quantities like pressure, density, energy, temperature, and concentration should always be positive, but numerical methods sometimes compute negative values during the solution process. This ...
862 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 ...
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### 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 ...
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### What are possible methods to solve compressible Euler equations

I would like to write my own solver for compressible Euler equations, and most importantly I want it to work robustly in all situations. I would like it to be FE based (DG is ok). What are the ...
375 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 ...
519 views

### Which time-integration methods should we use for hyperbolic PDEs?

If we employ the Method of Lines for discretization (separate time and space discretization) of hyperbolic PDEs we obtain after spatial discretization by our favorite numerical method (fx. Finite ...
770 views

### Finite difference scheme for "wave equation", method of characteristics

Consider the following problem $$W_{uv} = F$$ where the forcing term can depend on $u,v$ (see Edit 1 below for the formulation), and $W$ and its first derivatives. This is a 1+1 dimensional wave ...
704 views

### Coupling FEM DG methods to Riemann solvers

Are there any good papers and or codes that couple discontinuous galerkin finite element solvers with Riemann solvers? I need to explore coupling elliptic and hyperbolic problems but most splitting ...
543 views

### Nonlinear wave equation - Finite element or finite difference

I would like to know the which is more advantageous when it comes to solving nonlinear hyperbolic equations, Finite Element or Finite difference methods? Which method will be better in capturing ...
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### Numerical quadrature in Discontinuous Galerkin

I would like to know which is the best way to integrate numerically Legendre polynomials. I am building up a Discontinuous Galerkin CFD code for which Legendre polynomials are used as basis functions ...
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### Numerical methods for the continuity equation with Sobolev vector field

Consider the continuity equation $$\partial_t \rho(x,t) + \operatorname{div} (b(x,t) \rho(x,t)) = 0, \qquad t \in [0,T], \quad x \in \mathbb R^N,$$ with $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
244 views

### Stability analysis for a hyperbolic PDE on staggered grid

I am trying to understand the stability of a finite difference equation on the staggered grid. I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...
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### Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
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### How to derive the stability of time stepping schemes?

This is more of a mathematical question but since we deal with this all the time in computational science, maybe it is relevant in this forum too. I am an engineer and I am learning how to model the ...
817 views

### CFL condition for variable coefficients

I understand that the constant in the Courant-Friedrichs-Lewy condition is defined as $\mathrm{CFL} = \frac{u \Delta t}{\Delta x}$, where $u$ is the principal coefficient. I came across this post: ...
303 views

### mathematical statement of "open" boundary condition

For your information, the original equation comes from here. Note: You DON'T have to read the paper. I will make the question as self-contained as possible. The central equation to solve is equation (...
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### How to set up a shock tube problem such that the solution includes a shock with a specified Mach number

One of the famous and convenient test cases for shock wave modeling is the 1D Sod's shock tube. This is a Riemann problem for the compressible Euler equations of gas dynamics. The initial set up has ...
2k views

### amplification factor for the Crank Nicolson scheme for the advection equation

I will try one more time being more detailed and careful. Consider the transport equation of the form $$u_t+au_x=0, t\in[0,T],x\in \mathbb{R}, a>0$$ and initial condition $u(0,x)=u_0(x)$. I would ...
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### Measure the convergence rate of a discretization of a wave equation

I'm currently trying to approximate the following type of wave equation (in weak formulation): Let $\Omega \subset \mathbb{R}^d$ ($d=2$) be some polygonal domain. We seek a function \$u \in L^2\left(0,...
147 views

### High order unconditionally stable discretization for a scalar hyperbolic PDE

In order to numerically solve the following differential equation: \text{Fr}\{f\} := v(k)\dfrac{\partial f(z,k)}{\partial z} - F(z) \dfrac{\partial f(z,k)}{\partial k} = -\dfrac{f-...