# Questions tagged [hyperbolic-pde]

Hyperbolic partial differential equations describe wave behavior.

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### 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 ...
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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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### 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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### what do positive real parts of eigenvalues mean?

I am solving a 1D advection problem of the the form $$d{Q}/dt = [A]{Q}$$ where {Q} is the vector of unknowns and [A] is the matrix of coefficients of spatial discretisation. I have worked out the ...
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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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### 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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### 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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### 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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### Unwanted Oscillation in FDM simulation of elastic wave equation

I am using staggered grid FDTD for solving elastic wave equation. A description of which can be found at (geodynamics.usc.edu/~becker/teaching/557/reading/Virieux1987.pdf). I have generated a ...
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### What is the best option in terms of library or software to solve this system of hyperbolic PDEs?

I want to solve a system of coupled nonlinear 1-D PDE $(\partial_{tt} + \alpha\partial_t)u_i(x,t)=\partial_{xx}(\sum_{j=1}^{j<i}ju_j(x,t)+i\sum_{j=i}^{n}u_j(x,t))-\sin(u_i(x,t))+f$, using method of ...
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### Implicit time integrator for Chebyshev collocation method for linear hyperbolic system

I want to solve linear hyperbolic system using Chebyshev collocation method. As this method puts severe constraint on the time step for the explicit time integration, I decided to switch to implicit ...
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### Conditions for always positive gradient of heat field in evolutionary thermo-elastic system

I am investigating stability and convergence of series of approximations for coupled thermoelasticity problem yielded by one-step recurrent time-integration scheme. I've managed to show that the one-...
I am trying to solve the following 1D inhomogeneous wave equation. Forgive me if I a miss any rigorous mathematical concept. $$\frac{\partial^2 u}{\partial x^2} - \frac{1}{c^2}\frac{\partial^2 u}{\... 0answers 166 views ### Enforcing continuity conditions in pseudospectral domain decomposition methods for time dependent PDEs I have a partial differential equation of the form$$ \frac{d}{dt}f(x,t) = \Theta(x) f(x,t) \qquad \Theta(x) \sim \left[\frac{d^2}{dx^2} + k^2(x)\right] $$subject to f(x,t=0) = f_0(x), and f(x=0,t)... 0answers 73 views ### Transient advection equation with stabilized FEM I am interested in solving the transient advection equation \left\{\begin{array}{ll}\partial_{t} u+\beta \cdot \nabla u=f & \text { in } \Omega, t>0 \\ u=0 & \text { on } \partial \Omega^{-... 0answers 47 views ### Cauchy problem ill-posed? Find the solution to the Cauchy problem consisting of the wave equation :$$u_{xx}-u_{yy}=0$$together with initial conditions:$$ u(x,0)=0,u_{y}(x,0)=g(x)$$for some known initial datum g. Is ... 0answers 65 views ### discretizing advection equation with variable wave speed + stability I currently have a code that solves u_t+ cu_x=0 with periodic boundary conditions, and constant c (using an upwind method). I'm wondering how I would alter this code to solve something of the form ... 0answers 76 views ### Numerical scheme to calculate the normal mode of a set of hyperbolic PDEs? I would like to solve the linearised, ideal, MHD equations, where the gas pressure is zero.$$\frac{\partial u_x}{\partial t}=v_A^2(x,z)\left[\nabla_{||}b_x - \frac{\partial b_{||}}{\partial x}\right],...
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} \...