# Questions tagged [discretization]

The process of representing a continuum space with a finite set of points/elements

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### Dirichlet BCs - alternative implementation methods

I am having problems with solving a hyperbolic wave problem with Dirichlet BCs. I have tried reducing the time step sizes, which does not affect the results, and notices increasing the number of nodes ...
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### upwind schemes for solving inviscid euler equations

I'm new to the modelling of inviscid euler equations. I have come across few different upwind schemes that are used instead of central differencing schemes to model such flows, such as flux vector ...
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### Peculiar error when solving the Poisson equation on a non-uniform mesh (1D only) finite volume method

I have been trying to debug this error the last few days I wondered if anybody has advice on how to proceed. I am solving the Poisson equation for a step charge distribution (a common problem in ...
315 views

### Pseudo-inverse of a discretized operator with a null space?

Is there a way to understand what happens when a singular operator is discretized and inverted using the pseudoinverse (say using the SVD Moore-Penrose pseudoinverse)? For example, if we discretize ...
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### Should the Jacobian of a system of PDEs be calculated from the main equations of the discretised equation?

I am solving a coupled system of non-linear PDEs in 1D. Something like, $$u_t = F_1(u,v,w) \\ v_t = F_2(u,v,w) \\ w_t = F_3(u,v,w)$$ where each variable is a function of $x$ (the spatial dimension)...
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### Time stepping in comsol multiphysics

I would like to know which is the algorithm Comsol uses in order to correct the time step it uses. For example, when you try to solve an equation you've written in the PDE coefficient interface, and ...
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### Implementing an adaptive discretisation (upwind/central hybrid) for the advection-diffusion equation on a non-uniform mesh

I am following the approach of Hundsdorfer from Numerical solution of time-dependent advection-diffusion-reaction equations in which they introduce adaptive upwinding. The method can adapt from pure ...
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### How should non-constant coefficients be treated with finite-volume first order upwind scheme?

Starting with the advection equation in conservation form. $$u_t = (a(x)u)_x$$ where $a(x)$ is a velocity which depend on space, and $u$ is a concentration of a species which is conserved. ...
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### Improving the time integration of implicit discretized PDE with a non-linear source term

This might be a naive question, but when applying a implicit discretization to a PDE with a source term, should the source be averaged in time? For example if we take the diffusion equation with a non-...
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### Mehrstellenverfahren for Poisson?

I found this method in this book for solving the Poisson equation with error converging with $\mathcal{O}(h^4)$. However, when I try to implement it for the 1D equation $-u''(x)=f(x)$, it only ...
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### Clarification on interpolation equalities given by Briggs

Briggs, "A Multigrid Tutorial" (pg. 35) has the following expressed as 2-D interpolation: \begin{align*} v^h_{2i,2j} &= v_{i,j}^{2h}\\ v^h_{2i+1,2j} &= 0.5\cdot(v_{i,j}^{2h} + v_{i+1,j}^{2h})\\...
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### Is Crank-Nicolson a stable discretization scheme for Reaction-Diffusion-Advection (convection) equation?

I am not very familiar with the common discretization schemes for PDEs. I know that Crank-Nicolson is popular scheme for discretizing the diffusion equation. Is also a good choice for the advection ...
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### What are Implications of Commutative Diagrams?

This question may be too broad. But I really want to know some concrete explanations. I often find various commutations appear here and there, which concerns the application order of two operators. ...
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### Lax equivalence theorem for integro-differential equation

Can the Lax equivalence theorem (http://en.wikipedia.org/wiki/Lax_equivalence_theorem) be applied to the discretization of integro-differential equations, or does a similar theorem exist for them?
I have the following boundary value problem: $$-(\alpha u')' + \gamma u = f$$ in $\Omega = (a,b)$ with b.c. $u(a) = u(b) = 0$ and $\alpha > 0, \gamma ≥ 0$ and $f:(a,b) \to \Re$ The weak ...