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8

For a hyperbolic system of equations, you can write your equation as $$ \frac{\partial \mathbf{u}}{\partial t} + [\mathbf{A}] \frac{\partial \mathbf{u}}{\partial x} = 0$$ and then perform an eigendecomposition $\mathbf{A} = \mathbf{Q} \mathbf{\Lambda} \mathbf{Q}^{-1} $ where $ \mathbf{\Lambda} $ is a diagonal matrix of the eigenvalues, then defining $\...


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

Wave equations like this can be rewritten as a hyperbolic system of first-order conservation laws: $$q_t + \nabla \cdot F(q) = 0.$$ The stable time step for any explicit numerical discretization depends on the CFL number, which is proportional to the maximum wave speed appearing in the problem. That speed can be found by computing the eigenvalues of the ...


5

I'm going to guess that there isn't one. In the usual method of lines for $u_t +au_x=0$, you end up with a system of ODEs of the form $$ u_t = Au. $$ So the restriction on the time step comes from (1) requiring that eigenvalues of $A$ lie in the domain of stability of whatever time-stepping method you use (e.g., if $A$ is skew-symmetric, with imaginary ...


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

Your image of the numerical domain of dependence is correct. But try to also draw the analytical domain of dependence, maybe this could help you to better understand what is going on. Note that the analytic solution of $u_t+au_x=0$ is $u(t,x)=u_0(x-at)$. So the slope of the actual dependence is $a$. The CFL condition just says that the numerical dependence ...


5

It sounds as if you're running a time-dependent linear elasticity simulation, right? Most likely, you're running an "Explicit" time-stepping scheme, which means that all of your information at time $t_{i+1}$ is computed entirely using information from time $t_i$ (or $t_{i-1}$...). One consequence of such a scheme is that in order to be a stable time ...


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 ...


4

The most commonly used explicit ODE solver in structural analysis is the central difference method. Because it is explicit, the solution becomes unstable if the time step is larger than a so-called critical time step. The calculation of this critical time step is straightforward and can be found in many references (e.g. page 808 in this text by Bathe ) and ...


4

As far is I know, a proof for the general case has not been presented so far. The most detailed analysis has probably been performed by Krivodonova & Qin 2013 in this work. Quoting from the conclusion: We have derived a closed form expressions for the eigenvalues of the DG spatial discretization applied to the onedimensional linear advection equation ...


3

I was going to write a comment, but the equation seems to view better in answers.. I assume Von Neumann analysis is the proper approach to derive this equation, but a coordinate transformation from the cartesian CFL condition (I took from wikipedia) is not somehow equivalent? Specifically: \begin{equation} \Delta t \sum_{i=1}^3 \frac{u_i}{\Delta x_i} = ...


3

You assert that for your scheme to be stable you need $|λ|≤1$ which is correct, however you find that all values for $c=γν$ will give $|λ|>1$. This means that the discretization you have applied is unstable and there is no CFL condition for stability. Consider using upwind differencing or an implicit method.


3

Though your problem is in 3 dimensions, you can gain a lot of insight by considering the 1D case. Let's omit the y and z terms and concentrate on x and t. Then, your problem would be equivalent to the classic advection-diffusion equation with an explicit Forward-in-Time, Centered-in-Space (FTCS) finite difference method. Knowing the name of the problem ...


3

Essentially, the time dependent Stokes equation looks like the heat equation: $$ \frac{\partial u}{\partial t} - \nu\Delta u = f-\nabla p, $$ plus the incompressibility condition $\nabla \cdot u=0$ that for the current discussion is immaterial. Thus, the same considerations for time step choice apply as for the heat equation. Consequently, using an ...


2

Short Answer: There is no general result that would hold for all implicit schemes. The reason is that how your method behaves with respect to numerical diffusion depends on the specific combination of numerical time stepping scheme and spatial finite difference. However, note that if your discretisation is consistent with your PDE and your PDE does not have ...


2

This is one of those scenarios where assessing the convergence order of your scheme is difficult, because, as you explained, your time-step is "linked" to your spatial discretization. What you could do is manufacture an analytical problem without any spatial error. For example, in finite elements, if you are using P2 polynomials, a second order ...


2

Yes. That's all there is to the stability condition. Taking the material properties - shear modulus ($\mu$), bulk modulus ($\kappa$) and density ($\rho$) - into account, the global critical time step is evaluated as the minimum of the critical time step for each element ($\Delta t^e$) $\Delta t^e = CFL * h^e / c_{\kappa}$ where CFL is the Courant-Friedrichs-...


2

For the simplest case of linear advection, applying von Neumann stability analysis gives the necessary restriction on the time step size, known as the CFL condition: $$ \frac{c\Delta t}{\Delta x}\le 1 $$ For a nonlinear equation such as Burgers' equation, it is not possible to derive an expression for the necessary restriction on the time step size due to ...


2

In the case of an anisotropic material, the phase velocities of waves traveling through that material are determined by Christoffel's equation: \begin{equation} [\rho c^2\delta_{ij} - C_{ijkl}n_jn_l][u_k]=0 \end{equation} Here, $\rho$ is the density of the material, $c$ is the propagation velocity of the wave, $\delta$ is Kronecker's delta, $n_j$ is the unit ...


1

It is done in the standard way through von Neumann stability analysis. The CFL condition on the pseudo timestep is straight forward. The restrictions due to the viscosity and other terms are a little complicated, but they can still be obtained. In the following paper the authors workout the time-step restriction on the pseudo time for an explicit Euler ...


1

At least in one dimensional case for linear advection I have not experienced for finite difference methods the CFL restriction depending on the precision order as you write. Conditionally stable explicit in time well designed numerical schemes have typically the restriction $dt \le C h$, I have such experience (and proofs) with 2nd and 3rd order accurate ...


1

I will expand the answer provided by @DavidKetcheson. First the equations are rewritten as a hyperbolic system of first-order conservation laws: $$q_t + \nabla \cdot F(q) = 0$$ or $$q_t + A q_x + B q_y + C q_z = 0$$ Where $q$ is a state vector formed with the components of the stress tensor $(\sigma_{11}, \sigma_{22}, \sigma_{33}, \sigma_{12}, \sigma_{23}...


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