# Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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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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### 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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### 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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### 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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### 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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### 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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
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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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### 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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### 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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### 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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### 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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### 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 ...
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 $$v=v_0(1-\frac{\rho}{... 1answer 194 views ### implicit method (crank-Nicolson) I not understand the procedure [closed] I'm trying to understand the passage through this equation can be written for easily solved with the fortran alghorithm in particular i don't understood the meaning of L_x and L_xx ... what (-1,0,1) ... 2answers 399 views ### Computing Roe's average density for General Equation of State I am solving a 1d Shock tube problem of compressible fluid obeying Euler equations(Hyperbolic pde). I am trying to simulate it using Finite Volume Method using Roe's Scheme. Half of the tube contains ... 0answers 291 views ### Crank-Nicolson scheme in space for advection equation Consider the equation$$\frac{\partial}{\partial t}v(t,x)=\frac{\partial}{\partial x}v(t,x)$$, for t,x\in\mathbb{R}. I'd like to solve this equation forward in space and backward in time, ... 0answers 263 views ### Why can I not solve the negative advection equation (backwards in time)? Suppose we have the negative, inhomogeneous advection equation:$$\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})$$... 1answer 671 views ### Crank-Nicolson method for inhomogeneous advection equation Suppose we have the inhomogeneous advection equation$$\left(\frac{\partial}{\partial x}+\frac{1}{c}\frac{\partial}{\partial t}\right)u(t,x)=v(t,x)$$for u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R} (... 0answers 618 views ### How to implement outgoing wave boundary condition I am solving the one dimensional wave equation: 0=\Box\phi = -\partial_t^2\phi + \partial_r^2\phi , using a Crank-Nicolson finite difference scheme, in the domain r\in[0,R]. First, I define \xi\... 0answers 63 views ### Discretization of a multi-function term I'm trying to do discretization to the following system: \frac{{\partial \rho }}{{\partial t}} = - \frac{{\partial \rho }}{{\partial x}}u - \rho \frac{{\partial u}}{{\partial x}} \frac{{\partial ... 1answer 112 views ### If I discretize a PDE in space with WENO and in time with an implicit method, do I need to solve a nonlinear algebraic system at each time step? I am attempting to solve a nonlinear advection diffusion equation$$\frac{\partial u}{\partial t} = \frac{\partial}{\partial x}(\frac{\partial u}{\partial x} + u^2)$$with Robin boundary conditions ... 3answers 297 views ### When is it safe to ignore the diffusion term in an advection-diffusion equation? Given the one dimensional equation: \epsilon\frac{\partial^2u}{\partial x^2} +\frac{\partial u}{\partial x} = 0  with 0\le\epsilon \ll1 with boundary conditions u(0) = 0 and u(1) = 2, we ... 0answers 142 views ### Good numerical method for solving the Kadomtsev Petviashvili equations. Is there an analytical solution? I need to solve the Kadomtsev Petviashvili (KP) equations$$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0 $$where$$\lambda=\pm 1 \;.$$My questions ... 1answer 314 views ### Numerical quadrature in Discontinuous Galerkin I would like to know which is the best way to integrate numerically Legendre polynomials. I am building up a Discontinuous Galerkin CFD code for which Legendre polynomials are used as basis functions ... 0answers 49 views ### Degree of freedom for elastic wave propagation problem I am solving a elastodynamics (vector valued elastic wave) equation. I create the 2D mesh in Gmsh discretised into triangular elements of second order. Therefore, it is my understanding that the ... 2answers 8k views ### Understanding the Courant–Friedrichs–Lewy condition I understand these equations in particular can be solved easily without use of computational methods. Although right now I am concerned with trying to solve these equations using numerical integration ... 0answers 124 views ### What is a good algorithm to solve a discrete continuity equation in Cylindrical coordinates? The equation is: \partial f/\partial t + \nabla \cdot (v f) = 0 , \;\; f \in [0,1]  and v is a velocity known at every grid cell. A more precise constraint is that f is either 0 or 1 but ... 0answers 201 views ### Spherical Advection Discretization (boundary nodes) Consider the spherical advection problem: describing the conservation of a property u in a closed spherical domain.$$ \frac{\partial u}{\partial t}+\frac{1}{r^2}\frac{\partial }{\partial r}\left(r^...
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,...