Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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32 votes
4 answers
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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 ...
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25 votes
3 answers
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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 ...
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18 votes
4 answers
630 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 ...
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17 votes
1 answer
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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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17 votes
2 answers
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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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13 votes
1 answer
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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 ...
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13 votes
1 answer
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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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13 votes
2 answers
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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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10 votes
2 answers
784 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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9 votes
2 answers
10k 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 ...
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9 votes
2 answers
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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 ...
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8 votes
1 answer
556 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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8 votes
1 answer
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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 ...
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8 votes
1 answer
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Higher order Lax-Wendroff type scheme?

Suppose we want to solve a hyperbolic conservation law $u_t+f(u)_x=0$. I really like to use Lax-Wendroff, which reads $u_j^{n+1} = u_j^n -\frac{\Delta t}{\Delta x}(g(u_{j+1}^n,u_j^n)-g(u_j^n,u_{j-1}^...
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8 votes
1 answer
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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8 votes
1 answer
276 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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7 votes
1 answer
369 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 ...
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7 votes
2 answers
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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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7 votes
1 answer
190 views

How does positivity preservation fit into the implication chain from monotone to monotonicity preserving?

I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws: monotone $\Rightarrow$ $L^1$-contractive ...
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6 votes
1 answer
278 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 ...
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6 votes
1 answer
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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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6 votes
2 answers
323 views

Numerical flux and source term in FVM (Burger's like equation)

I'm trying to solve the following equation with FVM $$u_t + f(u)_x = g(u)$$ where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
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6 votes
0 answers
147 views

Good numerical method for solving the Kadomtsev Petviashvili equations. Is there an analytical solution?

I need to solve the Kadomtsev Petviashvili (KP) equations $$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0 $$ where $$\lambda=\pm 1 \;.$$ My questions ...
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5 votes
1 answer
288 views

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

I am used to solve parabolic or elliptic non-linear PDE and the common methods to tackle non-linearity are Picard's iteration and Newton's method. I am a bit confused by the way things are done with ...
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5 votes
2 answers
1k views

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

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...
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5 votes
1 answer
355 views

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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5 votes
1 answer
275 views

Numerical methods for solving a mixed type nonlinear PDE

What type of numerical methods are there to solve PDE of the sorts of: $$\begin{align} &f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))\\ &u(x,0)=G_1(x)\\ &\frac{\partial u(x,0)}...
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  • 181
5 votes
0 answers
82 views

2nd-order TVD criteria for flux-limiter

Consider a nonlinear hyperbolic conservation equation: $$ \partial_{t}u = -\partial_{x}f(u) $$ The spatial derivative of $f(u)$ may approximated after a spatial discretization by $x_{j}=j\Delta x$ $$ \...
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5 votes
0 answers
85 views

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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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5 votes
0 answers
256 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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5 votes
0 answers
120 views

Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \begin{equation} \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
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4 votes
4 answers
2k views

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 ...
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  • 347
4 votes
1 answer
987 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: ...
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4 votes
1 answer
314 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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  • 269
4 votes
1 answer
107 views

How can I check mass conservation when solving the advection equation using an upwind scheme?

My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background Consider ...
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4 votes
0 answers
74 views

Library for solving multidimensional (n > 3) hyperbolic PDE systems

Does there exist a library (in any programming language) for solving (numerically) systems of multidimensional first-order linear PDEs in the form $$\mathbf{u}_{t}+\hat{A}(\mathbf{x})\mathbf{u}_{\...
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3 votes
2 answers
158 views

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

I am solving a 1D advection problem of the the form $$dQ/dt=[A]Q$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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3 votes
2 answers
2k views

Hyperbolic Equation PDE (Python)

I'm trying to solve the following first order hyperbolic PDE problem using method of lines: Hyperbolic Equation: $u_t = -u_x$ with initial condition: $u(0,x) = 0, 0 < x < 1$ boundary condition: ...
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  • 215
3 votes
1 answer
2k views

How to obtain an implicit finite difference scheme for the wave equation?

Suppose I had the following problem: $U_{tt}=U_{xx}+U_{yy}$ in $\Omega=[0,1]\times[0,1]$ $U(x,y,0)=f(x,y)$ $U_{t}(x,y,0)=g(x,y)$ $U=0$ on $\partial \Omega$ I know that there is an explicit ...
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3 votes
2 answers
171 views

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

I'm looking for some methods that could be directly applied to the PDE $$ \frac{\partial u}{\partial t} + \frac{(u_x)^2}{2} = 0\tag{*} $$ without converting it by $v = u_x$ to the Hopf equation $$ \...
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  • 165
3 votes
2 answers
1k views

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 ...
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3 votes
1 answer
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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3 votes
1 answer
1k views

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

I am trying to solve the following coupled PDEs: $$C_e\frac{\partial u(x,t)}{\partial t} = k_{ed}\frac{\partial^2u(x,t)}{\partial^2x} - G_{el}(u(x,t) - v(x,t)) + S(x,t)$$ $$C_l\frac{\partial v(x,t)...
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3 votes
2 answers
447 views

upwind schemes for solving inviscid euler equations

I'm new to the modelling of inviscid euler equations. I have come across few different upwind schemes that are used instead of central differencing schemes to model such flows, such as flux vector ...
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  • 301
3 votes
2 answers
783 views

Entropy fix for godunov scheme

For non linear system of hyperbolic PDE, The finite volume methods work well (because of inherent conservation). Godunov scheme is a very elegant solution philosophy. For linear system, it is nothing ...
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3 votes
1 answer
98 views

Discretizing the viscous component in 1 - D Navier stokes compressive flow

I've been working on modelling the NS equations in order to simulate shock waves. The equations are set up on the form: \begin{equation} \frac{\partial U}{\partial t} + \frac{\partial F(U)}{\partial x}...
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  • 55
3 votes
1 answer
216 views

Slope limiting for discontinuous Galerkin (DG) method

I had a question regarding the implementation of the TVB limiter for the RKDG method by Cockburn. I have seen that some implementations of the DG method use normalized Legendre polynomials such that ...
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3 votes
2 answers
449 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:...
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3 votes
1 answer
1k views

Euler's equations for a tube with varying cross-section

$\def\pd{\partial}$ $\def\l{\left}\def\r{\right}$ $\def\mdot{{\dot{m}}}$ $\def\eps{\varepsilon}$ Consider a tube with longitudinal coordinate $x$ from $0$ to $l$ and varying cross-section $A(x)$. ...
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  • 206
3 votes
2 answers
220 views

Finite volume discretization of non-conservative linear hyperbolic equation

Problem. Consider the one-dimensional adjoint Euler equations for $(x,t) \in \Omega \times [0,T]$ with $\Omega \subset \mathbb{R}$ and $T > 0$ $$ \varphi_t + \Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)...
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