Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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2
votes
1answer
71 views

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}...
2
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1answer
96 views

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{...
25
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3answers
16k views

Recommendation for Finite Difference Method in Scientific Python

For a project I am working on (in hyperbolic PDEs) I would like to get some rough handle on the behavior by looking at some numerics. I am, however, not a very good programmer. Can you recommend ...
3
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0answers
73 views

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{\...
0
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0answers
73 views

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^{-...
1
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0answers
61 views

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 ...
0
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0answers
47 views

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 ...
0
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0answers
68 views

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 ...
5
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0answers
74 views

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 $x_{j}=j\Delta x$ $$ \...
1
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1answer
358 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 ...
0
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0answers
76 views

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],...
2
votes
1answer
177 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 ...
1
vote
1answer
84 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 ...
2
votes
1answer
109 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) \\ ...
6
votes
2answers
275 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
2answers
55 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). ...
1
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0answers
92 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 (...
2
votes
1answer
213 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 ...
2
votes
1answer
166 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 ...
2
votes
0answers
206 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 ...
2
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0answers
66 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: $...
3
votes
1answer
79 views

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 ...
0
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0answers
51 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} \...
1
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0answers
40 views

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-...
32
votes
4answers
2k views

Why is local conservation important when solving PDEs?

Engineers often insist on using locally conservative methods such as finite volume, conservative finite difference, or discontinuous Galerkin methods for solving PDEs. What can go wrong when using a ...
5
votes
0answers
84 views

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 $b \in L^1((0,T), W^{1,p}(\mathbb R^N))$. ...
2
votes
2answers
404 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 ...
2
votes
1answer
76 views

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?
3
votes
2answers
197 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)...
3
votes
1answer
916 views

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)...
4
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0answers
73 views

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}_{\...
5
votes
1answer
267 views

Numerical methods for solving a mixed type nonlinear PDE

What type of numerical methods are there to solve PDE of the sorts of: $$\begin{align} &f(x,t,u(x,t))u_{xx} - g(x,t,u(x,t))u_{tt} = F(x,t,u(x,t))\\ &u(x,0)=G_1(x)\\ &\frac{\partial u(x,0)}...
1
vote
1answer
95 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{...
2
votes
0answers
75 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} ...
2
votes
0answers
147 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 ...
2
votes
0answers
100 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 ...
1
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0answers
69 views

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 ...
3
votes
0answers
220 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 ...
1
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0answers
62 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}{...
0
votes
1answer
197 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) ...
1
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0answers
296 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, ...
1
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0answers
263 views

Why can I not solve the negative advection equation (backwards in time)?

Suppose we have the negative, inhomogeneous advection equation: $$\left(\frac{\partial}{\partial x}-\frac{1}{c}\frac{\partial}{\partial t}\right)v(t,x)=u(t,x)\qquad(t\in\mathbb{R}_{+},x\in\mathbb{R})$$...
1
vote
1answer
676 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 $u,v:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ (...
1
vote
1answer
174 views

Debugging an implemented numerical method: which term gives the drop in accuracy?

I have a numerical scheme to solve an hyperbolic system of equations of this type (this a more simple version just for clarity): $$\frac{\partial u}{\partial t}+u\frac{\partial u}{\partial x}+v\frac{\...
1
vote
0answers
639 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\...
1
vote
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 ...
1
vote
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 ...
1
vote
3answers
309 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 ...
6
votes
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 ...
5
votes
2answers
961 views

Advice on numerical solution for 2D hyperbolic PDE with zero flux boundary conditions

I would like to numerically solve a hyperbolic PDE of the form $\frac{\partial\theta_t}{\partial t}(x,y)+\frac{\partial\left[\theta_t \gamma_t^x\right]}{\partial x}(x,y)+\frac{\partial\left[\theta_t \...