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Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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37 views

Staggered grids applied to the Hyperbolic PDE

Let the PDE be defined as \begin{align} & \begin{cases} u_t = v_x \\ v_t = u_x \end{cases}\; \; -1 \lt x \lt 1, t \ge 0 \\ & u(-1,t) = u(1,t) = 0 \end{align} Initial conditions for $u$ and $...
2
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0answers
56 views

Efects from the boundary in advection equation [duplicate]

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} ...
2
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0answers
19 views

Neumann boundary conditions in the Maccormack scheme

I am trying to solve the viscous Burger equation $$ \frac{\partial u}{\partial t} + u\frac{\partial u}{\partial x} = \mu \frac{\partial^2 u}{\partial x^2} $$ with Neumann boundary conditions. I am ...
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0answers
30 views

Trapezoidal Method applied to the advection equation: find $A$ in $U'(t) = AU(t)$

I have the following method of lines discretization of the advection equation: $$ U_j^{n+1}=U_j^n - \frac{ak}{2h}(U_j^n - U_{j-1}^n +U_j^{n+1} -U_{j-1}^{n+1} ) $$ which arises from the system $U'(t) ...
2
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0answers
28 views

WENO methods: why the characteristic wise method resulting big errors?

I was doing my research/project using WENO as the limiter in finite volume methods to solve hyperbolic conservation law. I have no idea why the result in the characteristic wise method has a big error ...
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0answers
49 views

Equal area algorithm to find shock location

I am looking to solve 1D burgers equation with various random initial conditions. What is the best algorithm to find the exact solution? One method that is covered in literature is the equal area ...
3
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0answers
109 views

Finite Element Stabilization for Drift-Diffusion/Advection-Diffusion Equations

I've tried my best to look through the relevant suggested similar questions when posting this, and hopefully this contains enough new material to not be considered a duplicate. I'm currently trying ...
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0answers
32 views

Convergence rate assesment in space/time

I am solving a hyperbolic PDE (e.g. the shallow water equations) which depends upon $x$ and $t$. Typically, the overall convergence rate is calculated by comparing the numerical error in different ...
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0answers
37 views

Apply flux-limiter to nonlinear hyperbolic equation

I am trying to solve the LWH traffic flow equation, which is a nonlinear hyperbolic equation $$\frac{\partial \rho}{\partial t}+\frac{\partial (v\rho)}{\partial x}=0,$$ where $$v=v_0(1-\frac{\rho}{...
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1answer
67 views

implicit method (crank-Nicolson) I not understand the procedure [closed]

I'm trying to understand the passage through this equation can be written for easily solved with the fortran alghorithm in particular i don't understood the meaning of L_x and L_xx ... what (-1,0,1) ...
2
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1answer
155 views

Computing Roe's average density for General Equation of State

I am solving a 1d Shock tube problem of compressible fluid obeying Euler equations(Hyperbolic pde). I am trying to simulate it using Finite Volume Method using Roe's Scheme. Half of the tube contains ...
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0answers
148 views

Crank-Nicolson scheme in space for advection equation

Consider the equation $$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$, for $t,x\in\mathbb{R}$. I'd like to solve this equation forward in space and backward in time, ...
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0answers
121 views

Why can I not solve the negative advection equation (backwards in time)?

Suppose we have the negative, inhomogeneous advection equation: $$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
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1answer
302 views

Crank-Nicolson method for inhomogeneous advection equation

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 $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
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0answers
156 views

How to implement outgoing wave boundary condition

I am solving the one dimensional wave equation: $0=\Box\phi = -\partial_t^2\phi + \partial_r^2\phi ,$ using a Crank-Nicolson finite difference scheme, in the domain $r\in[0,R]$. First, I define $\xi\...
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0answers
62 views

Discretization of a multi-function term

I'm trying to do discretization to the following system: $\frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial \rho }}{{\partial x}}u - \rho \frac{{\partial u}}{{\partial x}}$ $\frac{{\partial ...
1
vote
1answer
87 views

If I discretize a PDE in space with WENO and in time with an implicit method, do I need to solve a nonlinear algebraic system at each time step?

I am attempting to solve a nonlinear advection diffusion equation $$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x} + u^2)$$ with Robin boundary conditions ...
1
vote
3answers
143 views

When is it safe to ignore the diffusion term in an advection-diffusion equation?

Given the one dimensional equation: $\epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0 $ with $0\le\epsilon \ll1$ with boundary conditions $u(0) = 0$ and $u(1) = 2$, we ...
7
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0answers
110 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 ...
6
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1answer
185 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 ...
1
vote
0answers
44 views

Degree of freedom for elastic wave propagation problem

I am solving a elastodynamics (vector valued elastic wave) equation. I create the 2D mesh in Gmsh discretised into triangular elements of second order. Therefore, it is my understanding that the ...
1
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0answers
68 views

Methods to approximate discretized derivatives in PDEs

When solving a general PDE such as $$ \frac {\partial ^2 E}{\partial t ^2} = \frac {\partial ^2 E}{\partial z ^2} - \frac {\partial E}{\partial z} $$ this equation can be solved by the method of ...
2
votes
2answers
1k 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 ...
1
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0answers
71 views

What is a good algorithm to solve a discrete continuity equation in Cylindrical coordinates?

The equation is: $\partial f/\partial t + \nabla \cdot (v f) = 0$ $, \;\; f \in [0,1] $ and $v$ is a velocity known at every grid cell. A more precise constraint is that $f$ is either 0 or 1 but ...
1
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0answers
133 views

Spherical Advection Discretization (boundary nodes)

Consider the spherical advection problem: describing the conservation of a property $u$ in a closed spherical domain. $$ \frac{\partial u}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^...
3
votes
1answer
197 views

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,...
4
votes
0answers
175 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 ...
0
votes
1answer
53 views

hyperbolic equation and characteristics

I am a little confused about the connection between variables for the plain advection pde: $$u_t+au_x=0$$ So initially I thought $x$ and $t$ are independent and $u$ is a function of those, but then we ...
4
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2answers
89 views

numerical approach for system of non-linear partial-ordinary differential equations

I am interested in the numerical solution of the following system of non-linear partial-differential algebraic equations, where the independent variables are $X$ and $T$, representing non-dimensional ...
5
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0answers
105 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} - \...
4
votes
1answer
132 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} = -\dfrac{f-...
1
vote
0answers
200 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 ...
1
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0answers
72 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 ...
3
votes
0answers
119 views

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 v}{\...
4
votes
0answers
54 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 ...
2
votes
0answers
49 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 t^2}(s,...
0
votes
2answers
453 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 ...
3
votes
2answers
711 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: $u(0,x) = 0, 0 < x < 1$ ...
2
votes
4answers
506 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 ...
3
votes
2answers
135 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 $$ \...
1
vote
1answer
82 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 ...
6
votes
1answer
225 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 ...
3
votes
2answers
145 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 ...
2
votes
0answers
123 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 ...
2
votes
2answers
160 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?
3
votes
2answers
247 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 ...
0
votes
1answer
112 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}= \...
2
votes
1answer
165 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 ...
4
votes
0answers
189 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)}...
3
votes
1answer
319 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: ...