Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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Solving Burger's equation using second-order central upwind

I am numerically solving a Burger's equation with second order semi-discrete central upwind $$u_t+\left(\frac{u^2}{2}\right)_x=0$$ with the initial condition $$u(x,0)=\frac{1}{4}+\frac{1}{2}\sin(\pi x)...
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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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2 votes
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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 vote
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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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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5 votes
1 answer
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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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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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1 answer
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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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1 answer
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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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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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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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3 votes
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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 ...
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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 vote
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 answers
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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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  • 219
2 votes
1 answer
237 views

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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2 votes
1 answer
126 views

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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2 votes
1 answer
132 views

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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1 vote
2 answers
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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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6 votes
2 answers
316 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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102 views

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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2 votes
1 answer
239 views

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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3 votes
1 answer
201 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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2 votes
0 answers
226 views

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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2 votes
0 answers
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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 answers
57 views

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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1 vote
0 answers
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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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  • 131
5 votes
0 answers
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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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3 votes
1 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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3 votes
2 answers
210 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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  • 165
2 votes
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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3 votes
1 answer
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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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4 votes
0 answers
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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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1 vote
1 answer
101 views

Grid dependence of a numerical model

Statement of the problem Suppose, we consider the following model $$ \begin{array}{l} (1)~\mathbf{u}_t + \mathbf{F(u)}_x = \mathbf{S}(\mathbf{u},\mathbf{w}), \\ (2)~\mathbf{w}_x = \mathbf{P}(\mathbf{...
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2 votes
0 answers
78 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} ...
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2 votes
0 answers
163 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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2 votes
0 answers
104 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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1 vote
0 answers
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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 ...
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  • 322
3 votes
0 answers
229 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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  • 311
1 vote
1 answer
98 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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0 votes
1 answer
205 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) ...
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2 votes
2 answers
456 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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