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13 votes
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Why do we solve non-linearity in hyperbolic PDEs that way?

The good thing about the conservative form is that this comprises multiple models, such as Shallow Water Equations, Euler Equations or traffic models. An essential feature of hyperbolic equations is ...
Dan Doe's user avatar
  • 1,083
12 votes
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Understanding the Courant–Friedrichs–Lewy condition

I have two extra points I would like to add to Wolfgang's answer. A formulation of the CFL condition that I find more useful than the classic formula is this: A necessary condition for the ...
David Wells's user avatar
9 votes

Understanding the Courant–Friedrichs–Lewy condition

You are correct: If you satisfy the CFL condition, then all that guarantees is that your scheme is stable, i.e., the numerical solution does not go to infinity. But the CFL condition says nothing ...
Wolfgang Bangerth's user avatar
8 votes

why not all conservation laws solved numerically by hyperbolic methods

The core difference between the two PDEs you presented is that the heat equation is parabolic while Burgers equation is hyperbolic. This means that for Burgers equation changes in the solution travel ...
Dan Doe's user avatar
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6 votes
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When is it safe to ignore the diffusion term in an advection-diffusion equation?

The stationary equation you show transports information from the right to the left via the advection term; it also diffuses slightly. If you switch off the diffusion term altogether, then you only ...
Wolfgang Bangerth's user avatar
6 votes

why not all conservation laws solved numerically by hyperbolic methods

Actually, the heat equation can be, and often is, solved by hyperbolic methods. Instead of writing $q=-u_x$, write $q_t=-(u_x+q)/\tau$. Instead of the heat flux becoming instantaneously equal to the ...
Philip Roe's user avatar
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5 votes
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Crank-Nicolson method for inhomogeneous advection equation

Your equation can be written in the following fashion (any spatial derivative approximation is valid), once space is discretised: $$\frac{1}{c}\frac{du_i}{dt}=-\left(\frac{\partial u}{\partial x}\...
HBR's user avatar
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5 votes
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Is this the correct way for solving coupled 1d PDEs using finite difference methods?

This doesn't answer your question directly but instead suggests an alternate strategy. In general, unless you are interested in experimenting with numerical methods, I recommend using an existing PDE ...
Bill Greene's user avatar
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5 votes
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How to implement Lax-Friedrich flux splitting with WENO scheme

I think you mean with Lax-Friedrichs the local Lax-Friedrichs or Rusanov scheme [1], [2] where the wave speeds (left & right-going) are given by the maximum eigenvalue $\lambda_\max := \max_i | \...
Dan Doe's user avatar
  • 1,083
4 votes

How to derive the stability of time stepping schemes?

Another way you can find stability is to assume first you are solving the following model problem for complex $\lambda$: $$\dot{x} = f(t,x) = \lambda x$$ with exact solution: $x(t) = x_0 e^{\lambda t}$...
spektr's user avatar
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4 votes
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Grid dependence of a numerical model

Your numerical solution is probably just getting more accurate as you increase the number of grid points. Do you know or have you tried to derive the analytic (exact) solution for this problem? By ...
Savithru's user avatar
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4 votes
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Shallow water equations (SWE): well-posed initial data for single travelling pulse

What you are looking for is known as a simple wave solution, in which the variation in the solution belongs entirely to one characteristic family. If you had a linear hyperbolic system, this would ...
David Ketcheson's user avatar
4 votes
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Slope limiting for discontinuous Galerkin (DG) method

The paper of Cockburn and Shu [1] explains this. If the solution is $$ u_h(x) = u_i + u_{xi} \phi(x), \qquad \phi(x) = \frac{x - x_i}{\Delta x/2} $$ Then the limiter is $$ u_{xi} = minmod(u_{xi}, u_i -...
cfdlab's user avatar
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4 votes

Problems with manufactured solutions for 1D inviscid burgers' equation

You simply have a bug in your code. The flux is $\frac{1}{2} u^2$ and not $\frac{1}{4} u^2$.
ConvexHull's user avatar
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4 votes

stability of a numercial scheme for a hyperbolic system?

It is worth making some additional points. What you set out is just one version of the Lax-Wendroff method. That scheme is unique in one space dimension but has several free parameters in two or three ...
Philip Roe's user avatar
  • 1,154
4 votes

Numerical solution of 2D wave equation using Fourier transform and finite differences

Correction in the expression It appears that complex iota $i$ has not been included in the exponents in the expression for the inverse Fourier transform. The correct expression is: $$ u(x,y,t) = \iint ...
G R Krishna Chand Avatar's user avatar
3 votes

public solvers for the time-dependent Schrödinger equation?

I found the following package https://www.pci.uni-heidelberg.de//tc/usr/mctdh/doc/ featuring the multiconfiguration time-dependent Hartree (MCTDH) method for distinguishable particles. The method is ...
Arnold Neumaier's user avatar
3 votes

Numerical quadrature in Discontinuous Galerkin

Which basis you use for your finite element space does not matter for quadrature in general. If you use, for example, polynomials of degree $k$ (whether the Lagrange basis, or the Legendre basis, or ...
Wolfgang Bangerth's user avatar
3 votes

When is it safe to ignore the diffusion term in an advection-diffusion equation?

In the time-dependent nonlinear case, if you drop the diffusive term then you have a nonlinear hyperbolic problem. Solutions will naturally generate singularities (discontinuities) in finite time. ...
David Ketcheson's user avatar
3 votes
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Numerical methods that can be written in flux conservative form

The key feature to a conservative method is simply that the changes due to the fluxes cancel out (i.e., the flux leaving one cell is entering another), so the total mass is constant. Using the form ...
David Ketcheson's user avatar
3 votes
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Discretizing the viscous component in 1 - D Navier stokes compressive flow

Consider a FV method as an approximation of the integral conservation law. Starting from the one-dimensional, scalar conservation equation \begin{equation} u_t + f(u,\nabla u)_x =0, \end{equation} ...
ConvexHull's user avatar
  • 1,286
3 votes
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DG method for solving Hyperbolic Partial Differential Equation with Dirichlet Boundary Conditions

The Lax-Friedrichs flux roughly approximates a situation where the propagation of information can occur in both directions. This is why on the right face at $x_{N+1}$ you need to also know $u^{N+1}_h$ ...
helloworld922's user avatar
3 votes
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Shock Capturing Methods for Shallow Water Equations

Educational resources The shallow water equations are one of the simplest and most important systems of hyperbolic PDEs, and are frequently used in textbooks as an introductory example. There are ...
David Ketcheson's user avatar
3 votes
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Can the Runge-Kuta algorithm help in reducing numerical dispersion and anisotropy when using the FDM to solve the 2D wave equation?

Question 1 The temporal accuracy can probably be improved by using a fourth order Runge-Kuta algorithm instead of a standard three-point central difference approximation for the second order time ...
lightxbulb's user avatar
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3 votes
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Why does the FDM give a correct solution to a PDE with a discontinuous initial condition?

Looking at the plots of the errors, it seems that there is not much difference between the oscillations that you have in this case with respect to those that you had in your previous post (Can this ...
Rigel's user avatar
  • 417
2 votes

Measure the convergence rate of a discretization of a wave equation

You are asking very complicated questions for which there are likely no answers that can be rigorously proven. If you go back for a second and ask the same question for the solution of the Laplace ...
Wolfgang Bangerth's user avatar
2 votes

High order unconditionally stable discretization for a scalar hyperbolic PDE

Your problem looks very much like "implicit upwind method for advection equation", let me comment it from this point of view. Let us think that $f=f(z,k,t)$, $f(z,k,0)=f_0(z,k)$ and we search for $f(...
Peter Frolkovič's user avatar
2 votes

Numerical flux and source term in FVM (Burger's like equation)

The time evolution equation in hand is $\frac{\partial}{\partial{t}}{u} = L_1(u) + L_2(u)$ where the operators in the RHS are $L_1 = -\frac{\partial}{\partial{x}}{f(u)}$ and $L_2 = g(u)$. The ...
Maxim Umansky's user avatar
2 votes
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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?

Generically, yes. Since you have a nonlinear PDE you will end up with a nonlinear algebraic system no matter what spatial discretization you use. With WENO you will have a more strongly nonlinear ...
David Ketcheson's user avatar
2 votes
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Crank-Nicholson scheme for transport equation

Your problem seems to be the implementation of the boundary conditions. Apart from that you seem to not compute existing variables efficiently. If I understand your initialization right, then $x_2=a+...
Lutz Lehmann's user avatar
  • 6,064

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