Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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complexity of flux limiter techniques

My question is not related to any particular problem, rather, I am looking at the equations of the form $$u_t+c(t,x)u_x=0$$ and attempt to solve it numerically. According to http://en.wikipedia.org/...
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2answers
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 ...
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1answer
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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1answer
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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1answer
349 views

amplification factor of some schemes for the transport equation [closed]

Any source of FDM schemes for the transport equation starts by explaining that explicit central differences for the equation of the type $u_t+au_x=0$ can cause oscillations and unconditionally ...
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1answer
554 views

ENO/WENO vs monotone Hermite interpolation

I have see the method PCHIP in matlab that implements the monotone Hermite interpolation method which was originally proposed by Carlson in 1980s. It seem to accomplish the goal of preventing the ...
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1answer
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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0answers
166 views

Enforcing continuity conditions in pseudospectral domain decomposition methods for time dependent PDEs

I have a partial differential equation of the form $$ \frac{d}{dt}f(x,t) = \Theta(x) f(x,t) \qquad \Theta(x) \sim \left[\frac{d^2}{dx^2} + k^2(x)\right] $$ subject to $f(x,t=0) = f_0(x)$, and $f(x=0,t)...
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1answer
230 views

interpolation combined with methods of characteristics can cause oscillations for the transport equation?

I would like to know about the effect of using a higher order interpolator for the methods of characteristics. I am solving $$u_t+a(x,t)u_x=0$$ with some nonsmooth initial data $u_0(x)$ by the method ...
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2answers
546 views

Numerical Green functions for a nonlinear wave equation

I am trying to put down some code to get numerically the solution of the following PDE: $$ \partial^2_t\phi-\partial^2_x\phi+\lambda\phi^3=\delta(x)\delta(t). $$ Of course, there are several ...
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2answers
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 ...
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1answer
2k views

Finite difference coordinate transformation for spherical polar coordinates

I have part of a problem that is described by the momentum conservation equation: $\frac{\partial \rho}{\partial t} + \frac{1}{\sin\theta} \frac{\partial}{\partial \theta}(\rho u \sin \theta) =0$ ...
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1answer
363 views

Numerical solution of hyperbolic PDEs with nonconvex flux

In some hyperbolic PDEs the flux is nonconvex. One example is equations in magnetohydrodynamics. What are the complications in the wave structures of such problems? What general precautions one should ...
8
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1answer
270 views

Adaptive mesh refinement with perfectly matched layers?

We have an adaptive mesh refinement (AMR) code for solving the elastic wave equation with frictional fault interfaces (based on Chombo for those that are interested). One of the things that we have ...
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2answers
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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1answer
1k views

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 ...
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1answer
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 ...
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1answer
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 ...

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