# Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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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 ...
1 vote
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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: $$u(1, t) = U(t) = \int_0^1 k(1-y) u(y, t)dy$$...
1 vote
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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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### 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 ...
1 vote
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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 ...
105 views

### 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 $\phi$ ...
1 vote
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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) ...
127 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 ...
1 vote
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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}...
1 vote
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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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1 vote
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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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1 vote
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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). ...
379 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 ...
1 vote
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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 (...
283 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 ...
278 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 ...
256 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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### 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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### 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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...