Questions tagged [hyperbolic-pde]
Hyperbolic partial differential equations describe wave behavior.
141
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Numerical algorithm to compute second order solution of non-linear advection with source
I would like to solve the following equation,
$$\frac{\partial f}{\partial t}+\frac{\partial}{\partial x}\left({\mathcal A}f\right)=S(x,t)$$
where $f$ is a function of $x$ and $t$, ${\mathcal A}$ is a ...
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1
answer
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Simulating First Order Hyperbolic PDE with Finite Difference Scheme
I am trying to simulate a hyperbolic PDE with some control given by the following:
$$u_t(x, t) = u_x(x, t) + \theta(x) u(0, t)$$
with boundary conditions:
$$u(1, t) = U(t) = \int_0^1 k(1-y) u(y, t)dy$$...
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2
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Nonlinear Hyperbolic PDEs: Known solutions
I would like to collect some test-problems for nonlinear hyperbolic PDEs (Euler Equations, Shallow Water Equations, Ideal MHD, Acoustic Perturbation, ...) for which analytical solutions are known.
A ...
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Solving a system of PDEs with an ODE
I want to solve the following system of equations which consists two PDEs and one ODE:
\begin{align}
\rho_t+v\rho_x &= 0; \newline
Y_t+vY_x &= 0 ;\newline
v_t &= -\frac{1}{(\...
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Solving simplified 1D plasma fluid equations with finite difference
The following two equations represent a simple model of a plasma where ions are immobile.
$$n\frac{\partial u}{\partial t}+nu\frac{\partial u}{\partial x}=n\frac{d\phi}{dx}-\theta\frac{\partial n}{\...
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How can I check mass conservation when solving the advection equation using an upwind scheme?
My question is how to keep track of the "mass" being advected out of a model domain, for the 1-D advection equation, and an upwind differencing scheme. Following is the background
Consider ...
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Numerical methods for $u_t+c u_x= \frac{-c}{x}u$?
I am looking for possible numerical methods to solve the PDE
$$u_t+c u_x= \frac{-c}{x}u$$
I am particularly interested in a Finite elements method, although I am also curious if you can expose some ...
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A Finite Element Method for a first order PDE?
I want to develop a finite element method to solve for $u(x,t)$ the PDE:
$$u_t+c u_x= \frac{-c}{x}u$$
where $c$ is a constant.
so I am trying the following ( as Rothe's method? ) :
Letting $k= t_n- ...
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1
answer
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Numerical solution of 2D wave equation using Fourier transform and finite differences
This is the $2$-dimensional wave equation
$$ u_{tt} = u_{xx} + u_{yy} $$
with initial condition $u(x,y,0)=f(x,y)$ and $u_{t}(x,y,0) = 0$.
The inverse Fourier transform used is
$$ u(x,y,t) = \iint \hat{...
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Can you describe the Galerkin numerical method to solve the wave equation?
How would you describe the Galerkin method to solving the 3D wave equation
$$u_{tt}= c^2\Delta u$$
to someone who wants to implement it immediately?
More precisely, we want to solve the Cauchy problem
...
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1
answer
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stability of a numercial scheme for a hyperbolic system?
This is related to my question here https://math.stackexchange.com/questions/4447383/lax-wendroff-scheme-stability-analysis-for-a-linear-system-of-conservation-laws .
Consider the numerical scheme ...
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WENO scheme on the advection of a fluid in a compressible porous media
I am working on reactive transport and I need to solve this advection equation:
\begin{align}
\frac{\partial (\phi C)}{\partial t} = - \nabla \cdot \big(\vec{q}(\phi) C\big)
\end{align}
with $\phi$ ...
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Semi-lagrangian method for compressible fluid (non divergence free)
I am looking for a semi-Lagrangian method for advection with a non divergence free velocity field.
The equation is
\begin{align}
\frac{\partial C(x,t)}{\partial t} &= - \nabla \cdot (\vec{v}(x,t) ...
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1
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Shock Capturing Methods for Shallow Water Equations
I am looking for some help finding a numerical solution to the shallow water equations:
$\partial_tu+\partial_x(u^2/2+g\eta)=0$
$\partial_t \eta+\partial_x(u\eta)=0$.
where $u$ is the depth averaged ...
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1
answer
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How to implement Lax-Friedrich flux splitting with WENO scheme
I'm working on modeling a shock wave using the Euler equation with an advanced Equation of state and the fifth order WENO scheme. The equation are set up on the form:
\begin{equation}
\frac{\partial U}...
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1
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0
answers
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Chemical advection of a fluid in a porous media
I am trying to solve an equation for chemical advection of a fluid in a porous media with this equation:
$\begin{align}
\frac{\partial (\phi(x,t) C(x,t))}{\partial t} = - \nabla . (\vec{q(\phi, t, x)} ...
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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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TVD slope / flux limiters formulation
Even if the formulation is the same the TVD slope limiter can be applied:
to state reconstruction at the interface, in 1D FV formulation, we reconstruct the $Q^*_{j+1/2}$ and the $Q^*_{j-1/2}$ in the ...
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1
answer
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How to Impose nonhomogeneous Neumann Boundary Condition in the DG Formulation
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
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DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions
Consider the following partial differential equation
\begin{align}
\frac{\partial u}{\partial t}+\frac{\partial f}{\partial x} &= g(x,t), \ \ x\in \Omega = [x_{L},x_{R}] \\
u(x,0) &= u_{0}(x) ...
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How do I identify negative group speeds?
This question is a continuation of one of my other questions.
I've been trying to show that collocated (non-staggered) grids can suffer from negative group speeds in the linearized shallow water ...
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votes
1
answer
124
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A simple wave for the linear shallow water equations
I'm looking for a simple, right-traveling wave for the linear shallow water equations (1D). My question: what are the initial conditions (velocity $U_0(x)$ and/or average water height, average ...
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Appropiate Artificial Boundary Conditions for the radial part of the Klein Gordon equation?
I am trying to simulate the following equation using FDTD
$ \left(- \partial^2_t + \partial^2_x + V(x) \right) \psi(x,t) =0 $
subjected to the initial conditions $\psi(x,0) = f(x),~ \partial_t \psi(x,...
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votes
1
answer
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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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2
votes
1
answer
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Discontinuous Galerkin order of convergence on arbirary refined mesh: step-12 deal.ii tutorial
I'm learning DG methods and in order to practice a little bit I'm using the deal.ii library. In particular, I'm looking at step-12, where they solve
$$\operatorname{div}(\beta u) = 0$$
$$u = g_D \text{...
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answers
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TVD Lax-Wendroff with non-constant velocity
I am dealing with a linear advection equation with a non-constant velocity, where I would like to apply a TVD Lax-Wendroff scheme in 1D.
The equation is the following:
\begin{equation}
\frac{\...
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Transient advection equation with stabilized FEM
I am interested in solving the transient advection equation
$\left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-...
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For a hyperbolic PDE, is there any proof that the BDF2 method is stable for integrating them?
I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs.
Say I have a hyperbolic equation as $\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$. This system is ...
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Cauchy problem ill-posed?
Find the solution to the Cauchy problem consisting of the wave equation :
$$u_{xx}-u_{yy}=0$$ together with initial conditions: $$ u(x,0)=0,$$ $$u_{y}(x,0)=g(x)$$ for some known initial datum $g$.
Is ...
user37062
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discretizing advection equation with variable wave speed + stability
I currently have a code that solves $u_t+ cu_x=0$ with periodic boundary conditions, and constant $c$ (using an upwind method). I'm wondering how I would alter this code to solve something of the form
...
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2nd-order TVD criteria for flux-limiter
Consider a nonlinear hyperbolic conservation equation:
$$
\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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1
answer
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Problems with manufactured solutions for 1D inviscid burgers' equation
I'm having an issue with the easiest example of a nonlinear 1D PDE, the (inviscid) burgers' equation:
$u_t + uu_x = 0,~~(1)$
which can be rewritten as some convection equation
$u_t + f(u)_x = 0$
with ...
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0
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Numerical scheme to calculate the normal mode of a set of hyperbolic PDEs?
I would like to solve the linearised, ideal, MHD equations, where the gas pressure is zero.
$$\frac{\partial u_x}{\partial t}=v_A^2(x,z)\left[\nabla_{||}b_x - \frac{\partial b_{||}}{\partial x}\right],...
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votes
1
answer
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Numerical methods that can be written in flux conservative form
I have a system of non-linear PDEs that I expect to have shocks as well as the appearance of Gibbs phenomena (spurious oscillations that form near the shock) for 2nd-order methods or higher. I have ...
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Shallow water equations (SWE): well-posed initial data for single travelling pulse
This question concerns the 1-dimensional (i.e. only one spatial dimension) shallow water equations (SWE) shown below and how to find initial conditions such that we obtain a travelling pulse/wave ...
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Crank-Nicholson scheme for transport equation
This is my attempt to find the approximate solution of the folowing transport equation $$\left\{\begin{array}{ll}
\partial_{t} u+\partial_{x} u= (x^2-x)t+x^3/3-x^2/2, & t \in(0,0.4), x \in(0,1) \\
...
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2
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What space-time points should a known coefficient function be evaluated at when using the Lax-Friedrichs scheme to solve the transport equation?
For a scalar quantity $u = u(x, t)$, I'm considering the transport equation
\begin{align}
u_t + au_x &= 0, \qquad{x\in[0, L], \ t\in[0, T]}
\\
u(0, t) &= u_{\text{in}},
\\
u(x, 0) &= f(x).
...
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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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0
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Linearize non-linear PDE with BCs to hyperbolic problem: How does linearization affect BCs?
I am working with the Shallow Water equations that is a system of non-linear PDEs that simulate water waves propagation on some domain, in my cases the $x$-axis. I have here a GIF showing the results (...
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1
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Order of Accuracy Measurements on 1D Advection Methods
I am trying to learn about basics of computational fluid dynamics, at the moment on the simple example of linear advection in 1D.
I am am currently testing the theoretical predictions of the order of ...
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1
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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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0
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Implementation of boundary conditions for 1D Euler equations
I'm trying to solve 1D Euler equations with gravity in spherical coordinates using a finite-difference TVD MacCormack method on a non-uniform grid of $N$ components, following the method provided in ...
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0
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Solving a simple Shallow Water model
Hi. I have a question at Mathematics and they suggested post here, once it's not common. I transcript as following. Many thanks
I need to solve with basic methods this simple Shallow Water Model:
$...
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0
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71
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Testing the SUPG method and other methods for hyperbolic equations
I am interesting in integrating the simple equation
$$
\frac{\partial \phi}{\partial t} + \mathbf{u}\cdot\nabla \phi = 0
$$
with a Dirichlet boundary condition at the influx boundary ($\mathbf{u} \...
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votes
0
answers
51
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Dealing with spurious oscillations in particle tracking methods
I work on modelling high intensity discharge xenon-filled lamps.
The model governing the discharge is quite complex and sadly includes fluid dynamics.
After some time, I managed to implement a finite-...
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votes
0
answers
91
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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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$.
...
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votes
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answer
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Non-parametric models as solutions to Partial Differential Equations
In the realm of scientific computing, there are a plethora of techniques developed to solve Partial Differential Equations (PDEs). Many of the popular methods are variants of common techniques such as ...
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2
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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 $T > 0$
$$ \varphi_t + \Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)...
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1
answer
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public solvers for the time-dependent Schrödinger equation?
Are there efficient public solvers for the time-dependent Schrödinger equation with time-independent Hamiltonian and 2 or 3 degrees of freedom?
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Is this the correct way for solving coupled 1d PDEs using finite difference methods?
I am trying to solve the following coupled PDEs:
$$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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