# Tag Info

7

Such problems (sometimes called lateral Cauchy problems) are in general not well-posed (meaning they either lack a solution, or there are infinitely many of them, or the solution is unstable under perturbations of the boundary conditions). For parabolic (or dissipative) equations, it makes sense to study the stationary limit (simply omit the term $u_t$ in ...

6

Let me give a part of your answer, I would need some more indications from your side to answer you fully. So please read, and write some comments so that I can complete my answer. About notations in numerical methods There are a few mistakes in the way you write your equations. These are only details, but for someone who is used to it, it can a bit ...

6

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 about how accurate the numerical solution is. For that, indeed $\Delta z$ and $\Delta t$ must also be small enough compared to the features of the exact solution. ...

6

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 have transport from the right to the left, and you need to also drop the boundary condition at the left: because information is from the right to the left, nothing ...

5

Eigenvalues with zero eigenvalue correspond to purely oscillatory modes. You can see it by diagonalising the system. Your matrix $A$ can be written as \begin{equation} A = P \Sigma P^{-1} \end{equation} where $\Sigma$ is a diagonal matrix with entries $\lambda_n$ corresponding to the eigenvalues of $Q$. You can now transform your ODE to \begin{equation} dQ/...

5

The CFL number (or simply Courant number) is defined locally: $\nu = u(x,t) \frac{\Delta t}{\Delta x}$. The velocity may vary in time and space and the grid may be non-uniform (in $x$ and/or $t$). So the CFL number can be different at each point in time and space. A necessary condition for convergence of a consistent method is that the CFL condition be ...

5

I have implemented the solution derived below using PyClaw in an IPython notebook. If you download that, you can adjust the initial values and see the computed solutions. General setup In the solution of a Riemann problem for the 1D Euler equations, there are generally 3 waves. Two of them are genuinely nonlinear; the other is a contact discontinuity, ...

5

First-order hyperbolic equations model conservation laws; as the alternative name "transport equations" suggests, they transport information along so-called "characteristic curves" with a finite speed of propagation. For the simple model equation $u_t + u_x = 0$, you can show (e.g., by separation of variables) that the solution should be of the form $$u(t,x)... 5 I have two extra points I would like to add to Wolfgang's answer. A formulation of the CFL condition that I find more useful than the classic formula is this: A necessary condition for the stability of a numerical scheme is that the numerical domain of dependence bounds the physical domain of dependence. This is exactly what good old$$ \dfrac{\Delta ...

5

Your equation can be written in the following fashion (any spatial derivative approximation is valid), once space is discretised: $$\frac{1}{c}\frac{du_i}{dt}=-\left(\frac{\partial u}{\partial x}\right)_i(t) + v_i(t) \tag{*}$$ Keep in mind that $v_i(t) = v(x_i,t)$. Now the system of equations depends only on time $t$ you can apply Crack Nicholson method to ...

4

In 1-D, the first discretization you've presented is correct. The matrix equation does not look right, though. For starters, it's not clear what your boundary conditions would be or how you would incorporate them. In $N$ dimensions, the advection equation looks like \begin{align} \frac{\partial{u}}{\partial{t}} + \sum_{i=1}^{N}\frac{\partial}{\partial{x^{i}...

4

The first two can be decomposed into a system of first-order advection equations of which there are a number of standard test problems often involving both smooth and discontinuous initial conditions and all of which can be solved analytically. For instance this solution is to the linearized acoustics equations. The choice between one or the other is ...

4

This is a Hamilton-Jacobi equation. You can read about how to apply WENO to such equations in Section 4 of Chi-Wang Shu's 2009 WENO review paper, and references therein.

4

To elaborate on Wolfgang's answer: since hyperbolic PDE semi-discretizations with centered differences have purely imaginary eigenvalues, they are only neutrally stable. For linear problems (e.g., the acoustic wave equation or linear Maxwell's equations) this is fine and such methods are commonly used. For instance, in electromagnetics centered differences ...

4

Central differencing schemes are not stable if you have advection dominated problems. There really was no other trivial  alternative to developing upwind schemes.  There are a few other stabilization methods, of course, but in the finite difference of finite volume context, almost everything that was developed over the first 30 years of numerical ...

4

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 looking at your plots, it seems like the exact solution has a shock (discontinuity) occurring at around x = 4.5, and the numerical method is resolving it with ...

4

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 solver instead of trying to write one yourself. This advice is particularly true for numerically-challenging problems, like yours appears to be. Since you are ...

4

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

3

One way is via von Neumann stability analysis. This technique hinges on the fact that the Fourier transform can sometimes be regarded as an $L^2$ isometry. This and other techniques such as direct norm estimates are discussed in, for example, the first few chapters of: Strikwerda, J. C. (2004). Finite difference schemes and partial differential equations (...

3

Do I definitely need to know the eigenvectors/eigenvalues of the system if I need to use an upwind scheme for such flow? No, it is certainly not necessary to use the full eigenstructure of the system, especially for lower order (order 1-2) schemes. Some of the most commonly used Riemann solvers are Lax-Friedrichs, HLL, HLLE, and HLLC (see Wikipedia). Lax-...

3

In the 2-D case, this corresponds to an elliptic fixed point (an orbital, I believe). You might look into Lyapunov stability, hopefully someone will be able to recommend a good resource on that. I'll try to have another look later to see if I can find resources on this. UPDATE For related material, you might look into limit cycles, or the Poincaré-...

3

I have not studied this particular system before, and I'm sure that someone who has could say much more than I will. I don't think there is any reason to expect that discretizing the equations in the form you have them will lead to highly accurate conservation of total energy. In this form, conservation of total energy depends on cancellation of certain ...

3

This is just a small addition to David Ketcheson's comments about solving numerically. If you want to obtain a numerical solution using MATLAB, Lawrence Shampine has written a MATLAB solver for hyperbolic PDEs that is designed to be (relatively) easy to use. If you look about half-way down this page: http://faculty.smu.edu/shampine/current.html you will ...

3

The answer to this question is that the system matrix $A$ of the linear equation system $$Au = b$$ that we obtain from a discretization scheme must be a M-matrix. This means in particular that $A$ is inverse monotone and this means that inverse matrix $A^{-1}$ is a monotone matrix. A monotone matrix $B\in\mathbb{R}^{n \times n}$ is a matrix, which has ...

3

$\def\rmC{{\mathrm C}}$ $\def\vr{{\vec r}}$ $\def\vA{{\vec A}}$ $\def\ve{{\vec e}}$ $\def\l{\left}\def\r{\right}$ In the following I try to show that Bale's form is correct. If you complain, leave a comment. The equations we can trust are the integral equations. Mass Balance: $$\l[\int_0^x \rho(x',t') A(x') dx'\r]_{t'=0}^t + \int_{t'=0}^t \l[(\rho u)(x'... 3 Many times, the equations to be solved represent a physical conservation law. For example, the Euler equations for fluid dynamics are representations of conservation of mass, momentum, and energy. Given that the underlying reality that we are modeling is conservative, it is advantageous to choose methods that are also conservative You can also see ... 3 Another way you can find stability is to assume first you are solving the following model problem for complex \lambda:$$\dot{x} = f(t,x) = \lambda x$$with exact solution: x(t) = x_0 e^{\lambda t} Then you define your scheme, let's start with Explicit Euler:$$ x_{k+1} = x_{k} + h \; f(t,x_{k}) $$where h is the step size. Based on this scheme, you ... 3 The problem is that odeint solves first-order ODEs of the form$$\dot u(t) = Au(t),$$but the wave equation is (after discretization in space) a second-order ODE of the form$$\ddot{u}(t) = Au(t). (And in fact, your plot looks quite reasonable for the solution to a parabolic equation.) To apply a black-box ODE solver (which is really not such a great idea,...

3

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 monomials, or anything else) then the integrand of the mass matrix has polynomial degree $2k$, and you need a Gauss quadrature with $k+1$ points to integrate ...

3

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. To extend the solution beyond that time, one must consider weak solutions, and uniqueness is lost. To specify a unique, physically relevant solution one ...

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