12

No, positive definiteness (and symmetry) are only precondition to using the Conjugate Gradient method. But there are plenty of other iterative methods such as MinRes and GMRES that can be used for indefinite and non-symmetric matrices.


7

Yes. The constants that appear in the interpolation estimates upon which finite element error estimates are based contain minimum and maximum angles of triangles/tetrahedra (or similar geometric measures for quadrilaterals/hexahedra). These constants are smallest whenever you have equilateral triangles or square quadrilaterals.


4

When dealing with conservation laws like your case, you can often make use of the divergence theorem (as you did). You can then express the fact that the total mass within your integration region is preserved by the following surface integral: $$\oint_{\partial \Omega} k \nabla T \cdot \mathbf{n} ~\partial S = 0$$ Now, as it stands, it is irrelevant which ...


3

The correct way is to average the source term which you can do easily in this case as you have a polynomial. In general you can do the average with a quadrature. For second order accuracy if the source term is smooth as in your case, you can also just evaluate it at the cell center, which is like mid-point quadrature. It should not matter much in your ...


3

Generally the step from compressible Euler equations to the Navier-Stokes equations is not that hard, at least the coding part. If you want to implement it with an explicit scheme you have to consider the severe time step restriction of the parabolic contributions. One tricky part, at least for a consistent FV implementation, is the calculation of the ...


3

From the discussions and the paper, OpenFOAM seems to have implemented a measure of skewness. This answer is not an explanation why the different definitions of skewness might be equivalent, I am just going to justify why this is a measure of skewness. Consider following two elements -for sake of simplicity- Blue arrow is the outward surface normal fAreas[...


2

Solving the Riemann problem at the interface gives you the values of the solution at this interface just after $t_n$. Then, the basic assumption is that this interface state remains constant during the whole time step, which is true for piece-wise constant reconstruction (i.e. the solution is assumed uniform in each cell), and as long as the CFL condition is ...


2

Normal direction depends on the cell that you are writing equation for. the word outward is relative to the cell under study. In order to write equation for each of cells, i.e. $\Sigma \nabla T.n S_f=0$, stick to this : $\nabla T_{face}=\frac{T_c-T_i}{r_c-r_i}$ and assume $n$ as outward pointing normal vector for that face. I think your problem is that you ...


2

Most of this was already discussed in the comments, but I would like to elaborate and put a detailed answer. There are no elementary characteristics (definiteness, symmetry, bandwidth) which can tell you whether the underlying (mixed or not) FEM/FVM is stable to solve the continuous problem. You can not tell anything about that just by looking at those ...


2

An extended answer. For more arbitrary meshes you have to consider that generally CFD/FEM solvers rely on generic data-structures with element and side lists: Element list Side list Consider the following pictures, which is the standard case for simple Cartesian meshes. Since there is a single plus and a single minus side on each face, the definition is ...


2

Upwinding is not necessary. The best known and most-used limited central scheme is Nessyahu-Tadmor Nessyahu, H. and Tadmor, E., 1990. Non-oscillatory central differencing for hyperbolic conservation laws. Journal of computational physics, 87(2), pp.408-463. However a price is paid in that the contact discontinuity is badly smeared. Google Scholar will turn ...


2

You can use discontinuous Galerkin methods also for $P_0$ elements. It's true that the gradient in the cell interior is zero, so your formulation will exclusively consist of the jump terms at cell interfaces.


2

If $(x_i,y_i)$ is the centroid of the triangle, then $$ \frac{1}{\Delta_i}\int_{\Delta_i}{p(x,y)}\,dx\,dy = p(x_i,y_i) = \bar{\psi}_i $$ This is mid-point quadrature, which is exact for an affine function. The centroid of a triangle is the arithmetic average of its three vertices. Note that $$ x_i = \frac{1}{\Delta_i} \int_{\Delta_i} x dx dy = \frac{1}{3}\...


2

It depends on the problem being solved and the element formulation. The orthogonality criterion you state may not be as important as the shape of each element. For example in structural mechanics, with an irregular (e.g. automatically generated and refined) mesh and isoparametric elements, an important criterion for accuracy is the largest angle between ...


2

I would leave out a few things to make it more simple. This is how we do it for our code which is capable of using polyhedral meshes: https://github.com/nikola-m/freeCappuccino-dev/blob/master/src/mesh/geometry.f90 It is so called face based data structure. We use divergence theorem to compute geometrical data like volumes and cell center coordinates. This ...


1

The equation is derived from momentum balance, so there is nothing wrong with. But you can resolve the apparent paradox like this. Take a square control volume $C = [a,b] \times [c,d]$ and look at x-momentum balance $$ \frac{d}{dt}\int_C \rho u dx dy = -\int_c^d (\rho u^2)(b,y) dy + \int_c^d (\rho u^2)(a,y) dy + \ldots $$ The first term on the right is the ...


1

The eigenvalues given correspond to the flux jacobians in the Euler equations. Physically, they represent the speed at which waves of information can travel in a space-time domain. The goal is to compute the eigenvalues at each face based on the values in each neighboring cell so there is no 'left' or 'right' face in this equation, but instead a left and ...


1

The smooth solution turned out to have BC's applied in the following way: Walls and inlet: $\frac{\partial p}{\partial n}=0$ Outlet: $p=0$


1

Look at the diffusion dispersion relations: Upwind, ENO, WENO, see Fig. 2 DG, CG, see Fig. 15 Regards


1

Use a finite volume method. Define $$ \delta_x \phi(x,y) = \frac{\phi(x+\Delta x/2,y) - \phi(x-\Delta x/2,y)}{\Delta x} $$ $$ a_x \phi(x,y) = \frac{\phi(x+\Delta x/2,y) + \phi(x-\Delta x/2,y)}{2} $$ etc. For example, consider $\tau_{xx}$ which is required at $(i+1/2,j)$. $$ (\tau_{xx})_{i+1/2,j} = \mu_{i+1/2,j} \left[ \frac{4}{3} \delta_x u_{i+1/2,j} - \...


1

Just open up the parentheses, e.g., $\partial_{x} (\alpha \partial_{x} v) = (\partial_{x} \alpha) (\partial_x v) + \alpha \partial^2_x v$, where $\alpha=\mu$ or $\alpha=\mu u$ etc., and apply your central differences: $(\partial_{x} \alpha) (\partial_x v) = (\alpha_{i+1}-\alpha_{i-1})(v_{i+1}-v_{i-1})/(4h^2)$; $\alpha \partial^2_x v = \alpha_i (v_{i+1}+v_{i-...


1

Have you plotted the solution using a visualization package or just a cut through the middle of your domain? Sometimes just looking at the solution you're generating will give you an insight into where it's arising from. Based on your description, I'd say there's a mistake in your implementation of your boundary condition. ETA: The last sentence is probably ...


1

Disclaimer: I'm not 100% sure but I thought that I should provide my working solution to the above problem, for any future visitor that might have the same questions. The answer is for a steady temperature diffusion problem. What if the face is a boundary face and insulated? how to get $\phi_f$ in such case? Since a boundary face is in contact with only ...


1

To expand on Wolfgang Bangerth's answer, I think P0 DG schemes reduce to two-point cell-centered finite volume schemes. I don't know if DG convergence analysis always includes $p = 0$, but the resulting finite volume schemes can be shown to converge under appropriate "mesh orthogonality" conditions. https://math.unice.fr/~minjeaud/Donnees/...


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