# Tagged Questions

Hyperbolic partial differential equations describe wave behavior.

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### Stability analysis for a hyperbolic PDE on staggered grid

I am trying to understand the stability of a finite difference equation on the staggered grid. I could understand the Von Neumann stability analysis for the collocated grid for a simple acoustic ...
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### hyperbolic equation and characteristics

I am a little confused about the connection between variables for the plain advection pde: $$u_t+au_x=0$$ So initially I thought $x$ and $t$ are independent and $u$ is a function of those, but then we ...
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### numerical approach for system of non-linear partial-ordinary differential equations

I am interested in the numerical solution of the following system of non-linear partial-differential algebraic equations, where the independent variables are $X$ and $T$, representing non-dimensional ...
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### Using entropy functions for increasing numerical stability

Regarding the numerical stabilization of two-dimensional advection equation, \dfrac{\partial f}{\partial t} + \Big(\dfrac{d\varepsilon_1(k)}{dk}\Big)\dfrac{\partial f}{\partial z} - \...
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### High order unconditionally stable discretization for a scalar hyperbolic PDE

In order to numerically solve the following differential equation: \text{Fr}\{f\} := v(k)\dfrac{\partial f(z,k)}{\partial z} - F(z) \dfrac{\partial f(z,k)}{\partial k} = -\dfrac{f-...
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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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### Wave Equation PDE [closed]

I'm trying to solve the following PDE wave equation using method of lines: Wave Equation: u_tt = u_xx with initial condition: u(0,x) = sin*pi,u_t(0,x)=0, 0 < x < 1 boundary ...
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### Hyperbolic Equation PDE (Python)

I'm trying to solve the following first order hyperbolic PDE problem using method of lines: Hyperbolic Equation: $u_t = -u_x$ with initial condition: $u(0,x) = 0, 0 < x < 1$ ...
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### How to derive the stability of time stepping schemes?

This is more of a mathematical question but since we deal with this all the time in computational science, maybe it is relevant in this forum too. I am an engineer and I am learning how to model the ...
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### FVM - virtual node discretisation

I have come across the paper titled: A monolithic fluid structure interaction algorithm ... For a 1D grid, at the boundaries the paper uses virtual nodes $x_{0}$ and $x_{N+1}$ (page 372) and for ...
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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-...
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### Shallow Water Equations Boundary Conditions

I am trying to solve shallow water equations using DG methods. Flow over a bump is a common problem that comes up in this context. For example (http://loki.udc.es/info/asignaturas/calculo_ii/Finite%...
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### What are the good testing problems for hyperbolic equation?

I read the whole list of this question: Where can one obtain good data sets/test problems for testing algorithms/routines? But the answers are in different areas and I want to ask a specific area. I ...
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### Euler Equation Eigensystem with Gravity in the Energy Flux

I am modifying a conservative form of the Euler equations with gravity in the energy flux (see previous question: Energy Conservation in Conservation Laws with Source Terms) for use in a Riemann ...
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### Conservative finite-difference expression for the advection equation

Following on from the earlier question I am trying to derive a finite-difference scheme for the advection equation which is conservative. It was suggested that for advection equation with variable ...
I know from "Numerical Methods for Conservation Laws" by Randall J. LeVeque that there is an implication chain of properties of methods for conservation laws: monotone $\Rightarrow$ $L^1$-contractive ...