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The fundamental quantity in transport is the flux, $\mathbf v u$ for advection. The divergence theorem states that $$\int_\Omega \nabla\cdot (\mathbf v u) = \int_{\partial \Omega} (\mathbf v u) \cdot \mathbf n . $$ An equation is conservative when it is preserves this equality. Dropping to 1D with $\Omega = (a,b)$ and using the equation $u_t + (\mathbf v ...


11

This is expected behavior with the Verlet algorithm. It is a symplectic integrator, which means that it will preserve quadratic invariants to within roundoff error -- thus the form planetary orbits is maintained quite accurately (I assume you mean to say that the orbits are elliptical, not circular). However, the method is still a numerical approximation ...


9

There are the geometric integrations written by Ernst Hairer & co: E. Hairer, C. Lubich and G. Wanner (2002): Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential equations. Springer Series in Comput. Math., vol. 31 E. Hairer and M. Hairer (2002): GniCodes - Matlab programs for geometric numerical integration for ...


6

In a certain sense, @Geoff Oxberry is correct in saying that stability and preservation of quadratic invariants are not directly related. For instance, there exist explicit methods that will preserve energy for your problem (and they are certainly not $A$-stable). However, in another sense there is a relation between the two. Well-posedness First, note ...


5

Stability and preservation of invariants are unrelated. Hairer does a nice job of proving how various methods preserve different kinds of invariants in his book Geometric Integration. Chapter III is the most relevant to your question, where he demonstrates, for example, that RK methods and linear multistep methods preserve linear first invariants.


4

The condition that has to be satisfied at the node point 2 is that the influx equals the outflux. Here, this will then be $$ v_{12}C_{12} = v_{24}C_{24}+v_{23}C_{23}, $$ but $C_{ij}$ is not the average concentration in each pipe: It is the concentration at end point 2 of the pipes. If the velocities are constant in time, then you can of course rewrite ...


3

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 you wrote for a standard conservative method, if we sum $u$ on a grid with $M$ cells, we have: $$\sum_{m=1}^M u^{n+1}_m = \sum_{m=1}^M u^n_m - \frac{\Delta t}{\...


3

Your description "the function $h(y,x)$ gives the proportion of the mass $u(y,t)$ at position $y$ that moves to position $x$ in space in a small unit of time" seems to indicate a slight misunderstanding of derivatives. Since the term on the left of your equation ($\partial_t u$) is a time derivative, the "small unit of time" must be infinitesimally small. ...


3

I know this topic well for some class of PDEs, so I try to give you a general answer with one example from applications I am familiar with. 1.) Conservative form of PDE - this notion is used when your PDE is derived from a (typically integral) form of some (physical) conservation laws and you write this PDE in the origin form. Very often you can rewrite the ...


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

Use a first order upwind (for the convection component) and a second order central difference (for the diffusion component). So the end result would be equivalent to discretising the equation, $$ \frac{\partial u}{\partial t} = \frac{\partial \boldsymbol{v}}{\partial x} + D\frac{\partial^2 u}{\partial x^2} $$ So using the $\theta$-method you will end up ...


2

A lot of this will depend on the magnitude of $D$ and mesh size $h$. For pure convection, upwinding works well; however, upwinding ends up too diffusive for $D \approx h$. If $h \ll D$, then upwinding should behave less diffusively, though only will $D=0$ will the upwind scheme show minimal numerical diffusivity. As $D$ increases relative to mesh size, an ...


2

The way this is usually proven in the finite element context is different, but many finite element schemes satisfy conservation properties. For example, if you think of the Stokes equations, as long as the pressure space contains the piecewise constant functions, then mass is conserved. Similar properties can often be shown for the mixed Laplace equation ...


2

Sometimes the energy equation drives the flow. Problems like natural convection are driven by temperature differences. The energy equation is frequently converted into a temperature advection-diffusion equation which can be coupled to the momentum equation through the Boussinesq approximation. If there is a heat source or variation in the temperature in the ...


2

I am interested to know the significance of the energy conservation laws when modelling fluids (or other materials). Energy conservation is usually explicitly modeled with a differential equation when modeling temperature or internal energy matters; the main examples I can think of are compressible flow applications and flows with thermic chemical reactions ...


2

Probably not. If you have inviscid Burgers equation then your discontinuous initial condition should (somewhere) stay discontinuous because there is no viscosity. If there is discontinuity in the solution then there is Godunov's order barrier theorem which limits your convergence order to 1... Be careful to use fine enough grid to prove any order of ...


2

Yes, mass conservation holds, because in fact you are solving some sort of a relativistic diffusion equation. Why: $$\partial_{t} u + \partial_{x} v = 0$$ and: $$\partial_{t} v + \frac{1}{\epsilon^{2}} \partial_{x} u = -\frac{1}{\epsilon^{2}} (v - f(u))$$ But from first equation: $$\partial_{tt} u + \partial_{xt} v = 0$$ and from the second one: $$\...


2

Most pdes coming from physics have a divergence structure $$ u_t + \nabla\cdot F =0 \qquad \textrm{in} \quad \Omega $$ Then for any arbitrary control volume $D \subset \Omega$, we have $$ \frac{d}{dt}\int_D u dx + \oint_{\partial D} F \cdot n ds = 0 $$ i.e., the total quantity inside $D$ changes due to fluxes on the boundary of $D$. A numerical method ...


2

Will it give discontinuous result in flux terms? Yes, the flux terms in general may be discontinuous. Is it because shock tube problem is transient and equation $\rho_1 u_1= \rho_2 u_2$ is steady? I believe the "governing equation" you are referring to with $\rho_1 u_1 = \rho_2 u_2$ is part of the Rankine-Hugoniot jump conditions. This form is only ...


1

It's clear that the mass of $v$ is not conserved in general. Just take $u(x,t=0) = 0$ and choose $f$ so that $f(0)=0$. Then $$\left. \frac{\partial v}{\partial t}\right|_{t=0} = -\frac{v}{\epsilon^2}.$$


1

You may want a method that works right down to the coordinate singularity at $r=0$. I will do the spherical case, but the cylindrical case is similar. We want to avoid ever dividing by $r$. To think of a Finite Volume method, consider the control volumes to be concentric shells. The inner and outer radii of shell i are $(r_i-\Delta r/2, r_i+\Delta r/2)$, ...


1

The numerical integration scheme might not be energy-preserving. The error term is often strictly positive, allowing for non-secular growth of the energy. There are some ways around this, such as conservative Runge-Kutta integrators.


1

Think of the collisions between particles or walls as being modeled by a potential energy term of the form: $$U(r) = \begin{cases} 0, & \text{if $r > r_c$}, \\ +\infty, & \text{if $r \le r_c$}, \end{cases}$$ where $r_c$ is the radius of a particle (or wall). You can see that the energy goes to infinity, compensating the attractive forces. ...


1

What is "u" in your advection-diffusion equation? If it represents the mass-fraction of a species then the total mass of that species will likely vary over time. Also, in this case the advection-diffusion equation itself is the continuity equation of that species. I also believe that the equation should have a negative sign in front of the first term in the ...


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