Hyperbolic partial differential equations describe wave behavior.

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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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1answer
121 views

High order unconditionally stable discretization for a scalar hyperbolic PDE

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

Unwanted Oscillation in FDM simulation of elastic wave equation

I am using staggered grid FDTD for solving elastic wave equation. A description of which can be found at (geodynamics.usc.edu/~becker/teaching/557/reading/Virieux1987.pdf). I have generated a ...
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57 views

What is the best option in terms of library or software to solve this system of hyperbolic PDEs?

I want to solve a system of coupled nonlinear 1-D PDE $(\partial_{tt} + \alpha\partial_t)u_i(x,t)=\partial_{xx}(\sum_{j=1}^{j<i}ju_j(x,t)+i\sum_{j=i}^{n}u_j(x,t))-\sin(u_i(x,t))+f$, using method of ...
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My discretization of a wave equation in first-order form does not give correct solutions. What should I do?

I haven't much experience with conservation laws, shocks, etc. After reformulating my wave equation to 1st order system (velocity-stress): $$ \frac{\partial v}{\partial t} + A \frac{\partial ...
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27 views

Solving compressible inviscid Euler equations with shockwaves in polar coordinates

For the past several weeks I was attempting to adapt Lax-Wendroff or some similar scheme for polar coordinates. The process was complicated due to me being unable to find step-by-step derivations of ...
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39 views

Resolving a stiff hyperbolic problem with Neumann boundary conditions

I am trying to numerically resolve the equation for an Euler-Bernoulli beam that is inextensible, unshearable, and subject to planar deformations: $$\rho I(s) \frac{\partial^2 \theta}{\partial ...
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2answers
165 views

Wave Equation PDE [closed]

I'm trying to solve the following PDE wave equation using method of lines: Wave Equation: u_tt = u_xx with initial condition: u(0,x) = sin*pi,u_t(0,x)=0, 0 < x < 1 boundary ...
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217 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: $u(0,x) = 0, 0 < x < 1$ ...
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4answers
155 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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2answers
99 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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1answer
64 views

solving a hyperbolic set of equations - upwind type method

I want to solve a set of hyperbolic equations (not the Euler equations) using an upwind type method. I am interested in using a first order upwind scheme and one that is not based on the method of ...
5
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1answer
173 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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2answers
141 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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106 views

what do positive real parts of eigenvalues mean?

I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ 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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2answers
113 views

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

Are upwind schemes such as Godunov type methods superior to central differencing schemes? Do the reasons include superiority in modelling hyperbolic problems with Dirichlet BC's?
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2answers
108 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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1answer
86 views

How to define fluxes for two dimensional convection-diffusion equation?

I want to solve the following differential equation using control volume approach on a Cartesian mesh: $$\frac{\partial T}{\partial t} + \frac{\partial T}{\partial x} + \frac{\partial T}{\partial y}= ...
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1answer
110 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 ...
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139 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 ...
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1answer
122 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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CFL Neccessary Condition

The theorem states that if a difference scheme converges then it necessarily satisfies the CFL condition. How can this be proved?
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40 views

Debugging an implemented numerical method: which term gives the drop in accuracy?

I have a numerical scheme to solve an hyperbolic system of equations of this type (this a more simple version just for clarity): $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial ...
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57 views

Implicit time integrator for Chebyshev collocation method for linear hyperbolic system

I want to solve linear hyperbolic system using Chebyshev collocation method. As this method puts severe constraint on the time step for the explicit time integration, I decided to switch to implicit ...
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102 views

Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the ...
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1answer
143 views

Shallow Water Equations Boundary Conditions

I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example ...
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1answer
72 views

What are the good testing problems for hyperbolic equation?

I read the whole list of this question: Where can one obtain good data sets/test problems for testing algorithms/routines? But the answers are in different areas and I want to ask a specific area. I ...
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134 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 ...
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2answers
268 views

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

I need to solve the following hyperbolic equation in x and phi co-ordinates $$\frac{\partial \left ( -s/f \right )}{\partial \varphi }+\frac{\partial \left ( 1/f \right )}{\partial x}=0$$ $$\varphi ...
3
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2answers
358 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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1answer
268 views

What is the exact formulation of compressible Euler equation of gas dynamics in polar coordinates with artificial diffusion in 2D?

The interested equation is advection-diffusion equation. One of the canonical example is Navier-Stokes equations. However, I would like to let the coefficient of diffusion constant goes to zero, ...
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1answer
235 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 ...
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2answers
328 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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2answers
172 views

WENO reconstruction of flux involving derivative terms

I have a set of modified compressible Euler equations that I would like to solve using a WENO method. The issue is that the modified flux function involves derivative and filtering terms and I'm not ...
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1answer
403 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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44 views

Numerically evaluate 1D inhomogeneous wave equation solution

I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$ \frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 ...
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1answer
301 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 ...
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1answer
166 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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1answer
429 views

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 ...
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2answers
478 views

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 ...
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1answer
254 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
701 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
237 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
429 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
168 views

energy norm for transport equation

I asked this question before but did not have any luck with an answer. It might be a student level question but I need to understand that with possibly some help. I am considering the hyperbolic ...
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1answer
374 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
151 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 ...
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1answer
146 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 ...
6
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2answers
403 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
608 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 ...