Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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why not all conservation laws solved numerically by hyperbolic methods

Heat and Burgers equations for example are both conservation laws $du/dt+dq/dx=0$, where $q=-u_x$ $du/dt+df/dx=0$, where $f=u^2$ The former is usually solved by common finite differences and finite ...
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Can the Runge-Kuta algorithm help in reducing numerical dispersion and anisotropy when using the FDM to solve the 2D wave equation? [closed]

I am currently studying the effects of group velocity on the finite difference solution of the wave equation. Most of what I learned is from this source. I understand that high frequency components in ...
Amilox Lex's user avatar
2 votes
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Can I reduce my simulation error with a staggered grid, postprocessing and compatibility equation feedback?

What I did Using the finite difference method, I solved with a certain amount of error the following system of hyperbolic partial differential equations in cylindrical coordinates (the problem is ...
Nikola Ristic's user avatar
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How should I calculate the residuals of my numerical PDE solution?

I solved numerically the following wave equation with a source problem: $$ u_{rr} + {1 \over r}u_r + u_{zz} = {1 \over c^2}u_{tt} + s(r,z,t) \tag 1 $$ $$ u_r(0,z,t) = 0 \tag 2$$ $$ u_r(r_{max},z,t) = ...
Nikola Ristic's user avatar
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Numerical solution for inviscid Burgers' equation seems to have no breaking time?

So I'm trying to use the Lax-Friedrichs method to solve the inviscid burgers' equation with initial condition $$u(x,0) = \sin(x)$$, using $$u_m^{n+1} = \frac{1}{2}(u_{m+1}^n + u_{m-1}^n) - \frac{\...
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3 votes
3 answers
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Discretization of 2D advection equation with non-constant speed

Suppose I have a 2D advection equation $$\frac{\partial \rho}{\partial t}=-\nabla\cdot(\vec{w}\rho)$$ with $\vec{w}=(u,v)$ non-constant and having zero divergence. I want to numerically solve this ...
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Which numerical method can I use to solve this system of hyperbolic PDEs?

Backround The mathematical model I am trying to numerically solve models wave propagation inside a cylinder with specific material properties suited for dynamic loading. The cylinder's upper base is ...
Nikola Ristic's user avatar
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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 ...
Sayan's user avatar
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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: $$u(1, t) = U(t) = \int_0^1 k(1-y) u(y, t)dy$$...
Luke Bhan's user avatar
1 vote
2 answers
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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}{(\...
user44016's user avatar
5 votes
1 answer
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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 ...
nicholaswogan's user avatar
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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 ...
NotaChoice's user avatar
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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- ...
NotaChoice's user avatar
2 votes
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{...
Redsbefall's user avatar
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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 ...
NotaChoice's user avatar
1 vote
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 ...
NotaChoice's user avatar
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1 answer
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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 $\phi$ ...
Iddingsite's user avatar
1 vote
0 answers
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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) ...
Iddingsite's user avatar
2 votes
1 answer
164 views

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 ...
FluidMan's user avatar
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}...
Twm1995's user avatar
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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)} ...
Iddingsite's user avatar
6 votes
1 answer
338 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 ...
Iddingsite's user avatar
1 vote
0 answers
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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 ...
albiremo's user avatar
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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) ...
TheComander's user avatar
2 votes
1 answer
178 views

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) ...
TheComander's user avatar
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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 ...
theWrongAlice's user avatar
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1 answer
155 views

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 ...
theWrongAlice's user avatar
3 votes
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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,...
noir1993's user avatar
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3 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}...
Twm1995's user avatar
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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{...
FEGirl's user avatar
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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{\...
Iddingsite's user avatar
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0 answers
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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^{-...
balborian's user avatar
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0 answers
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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 ...
jengmge's user avatar
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0 answers
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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 ...
lrs417's user avatar
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5 votes
0 answers
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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 $x_{j}=j\Delta x$ $$ \...
user8384493's user avatar
1 vote
1 answer
724 views

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 ...
slvnklvr's user avatar
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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],...
Peanutlex's user avatar
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2 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 ...
user8384493's user avatar
2 votes
1 answer
156 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 ...
SimpleProgrammer 's user avatar
2 votes
1 answer
213 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) \\ ...
Almendrof66's user avatar
1 vote
2 answers
58 views

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). ...
Dan Burrows's user avatar
6 votes
2 answers
424 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 ...
VoB's user avatar
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1 vote
0 answers
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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 (...
SimpleProgrammer 's user avatar
2 votes
1 answer
304 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 ...
mivkov's user avatar
  • 203
4 votes
1 answer
321 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 ...
NumericalKid's user avatar
2 votes
0 answers
269 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 ...
Andrea Caldiroli's user avatar
2 votes
0 answers
72 views

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: $...
Quiet_waters's user avatar