Philip Roe
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What are the conceptual differences between the finite element and finite volume method?
7 votes

The basic difference is simply the meaning to be attached to the results. FDM predicts point values of any aspect of the solution. Interpolation between these values is often left to the imagination ...

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What is the origin of the spurious oscillations in the Crank-Nicolson scheme?
5 votes

There is something very basic that you should know about hyperbolic problems. Consider the most basic example $\partial_tu+a\partial_xu=0$ with a numerical marching scheme of the form $$u_j^{n+1}=\...

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Do computational scientists typically also become domain experts?
4 votes

I dont think there is a universal answer. In a computer science or math department you might get caught up in oversimplified problems about which you can prove things but which lack the complexity of "...

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Advection equation with finite difference: importance of forward, backward or centered difference formula for the first derivative
4 votes

You stumbled across something very fundamental that is important to understand. For any PDE we can define for each point its "domain of dependence" This is the region of space/time that is able to ...

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Can we simulate compressible flows by simple direct explicit calculation, without solving systems of linear equations (such as Poisson eq)?
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4 votes

Physically, vorticity can only be created (as opposed to transported, stretched or intensified after being created) either by the appearance of a boundary layer on a solid surface, or through the ...

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How important is learning hardware/architecture for scientific computing?
3 votes

I want to support the response from @Pseudonym, who makes the point that not everyone in the team needs to contribute to every aspect of the project. Something related to consider is that you are ...

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Finite difference for mixed derivatives on nonuniform grid
3 votes

My old answer has been edited in a way that I cannot agree with, but AG took such care to type everything out that Ill start again. This is a very old difficulty and the best textbook is Strang and ...

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How to discretize the advection equation using the Crank-Nicolson method?
2 votes

User 03161 asserts that the Crank Nicolson method is not appropriate for advection problems, but boyfarrell provides a working code with results visualized in a movie. In fact they are both correct, ...

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Which finite difference better approximates $uu'$?
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2 votes

There is really no such thing as a good finite difference equivalent to an operator. In the earliest days of scientific computing, the thought was that each differential operator would be replaced by ...

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Is upwinding needed for slope limiter / flux limiter and numerical flux?
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2 votes

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 ...

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How is central difference scheme second-order accurate?
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2 votes

The expression with $g=1/2$ is second order if and only if f is the midpoint of P and N.The expression with $g\in[0,1]$ is second order if f is on PN and$fN/Pf=g$. If f is anywhere else you need to ...

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How to show that a problem is ill-posed
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2 votes

Your calculation is fine. You have discovered data $u(x,0)=\cos(kt)$ whose solution becomes unbounded. Now consider initial $v(x)$ having unit initial norm and combine them into initial data $$U(x,0)=...

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Conservation violation in axisymmetric Diffusion Equation
2 votes

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 ...

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discrete definitions of curl $\nabla \times F$?
2 votes

This is made a lot easier by introducing the Calculus of Finite Differences. If $u_{i,j}$ is grid function defined for integer $i,j$ then the y-differencing operator $$\delta_y u=u_{j+1}-u_j$$ is ...

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Calculating a limit as parameter goes to infinity
2 votes

This is a bit primitive, but has a good chance to work. Plot $I$ versus $1/t$. Fit a smooth curve and find the intercept. This works a lot better if you have some theory predicting the behavior of $I$...

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Why is FEM theory only ever written down for simplices?
1 votes

Most FM methods require that we can identify pairs of adjacent elements. This is easy with elements derived from a quadrilateral/hexahedral grid that can be mapped onto an array structure. To deal ...

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How can I numerically differentiate an unevenly sampled function?
1 votes

I once worked on this for a bit. It makes a big difference where your data comes from. Typically it will contain errors from experimental data, or is quoted only with finite precision. In such cases ...

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Why is the FVM traditionally used in CFD, and FEM in computational structures?
1 votes

The most fundamental reason is that (static) structural problems often resolve themselves into finding minimum energy configurations and this is easily translated into minimizing the energy of a FE ...

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Finite difference for mixed derivatives on nonuniform grid
1 votes

In your sketch, you are having 9 grid points so you could have fit all 9 values to a 2-D cubic polynomial. Then the Taylor series expansion becomes $(I)\qquad f(x+b,y+d)\approx f + b f_x+d f_y +\frac{...

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von Neumann stability analysis for spatial variable flux
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1 votes

Im not sure there is a rigorous justification, but consider that stability is defined (should be defined) in the following way. That the numerical solution remains bounded as the mesh sizes in time ...

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How to estimate the error of trapezoidal rule using discrete data?
1 votes

There is a graphical version of this method (Richardson extrapolation) that can be very insightful. Use least three values of $h$, not necessarily different by factors of two or any other simple ...

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How to determine if 2 rays intersect?
0 votes

Since any two non-parallel lines must intersect somewhere (according to Euclid) I imagine that the OP intended a slightly different question. For example, do the rays intersect within the convex hull ...

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How can I derive a bound on the spurious oscillations in the numerical solution of the 1D advection equation?
0 votes

One approach is via the equivalent equation, that is, the differential equation to which your discrete method gives the closest aproximation. This is never the differential equation that you intended ...

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Finite difference for 1D wave equation: why the spike initial data results in a noisy output?
0 votes

You can only expect to get a sensible answer if your data can be resolved by the grid. Shannons Sampling Theorem is central to all digital signal processing and states that no signal with wavelength ...

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In Matlab, how can I be consistent with units?
0 votes

In research and teaching, mostly use dimensionless equations. A neat way in aerodynamics is to put the free stream values as $p=1, U=M, \rho=\gamma$ (so that $a=1$) But if you are designing something ...

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Finite differences on domains with irregular boundaries
0 votes

This question opens a can of worms, as is made evident by the variety of answers given. Many of these make useful points, but a really helpful answer would take account of considerations that have not ...

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Role of the numerical flux in DG-FEM
0 votes

When you choose the test function equal to the trial function in the DG method you are creating an optimization problem. That is, you have a Galerkin rather than a Petrov-Galerkin method. You are ...

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Why does multiplying two first derivative finite difference matrices not give the matrix for the second derivative?
-1 votes

Your matrices are not well defined. The second derivative operator for example, takes three consecutive data values and outputs one second derivative value. That is not expressed by writing a square ...

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