# Questions tagged [numerics]

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### Research in Inverse Problem and Numerical PDE

I am taking a Thesis-based Master degree now and I am going to choose my supervisor soon. I plan to take a PHD degree after graduation, so if possible, I wish my PHD research area could be an ...
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### How do you numerically solve a multivariable ODE system with different time steps per state variable?

If you have a large multivariable ODE system, and certain processes occur at a much smaller time scale, how can you implement a solver that uses smaller time steps for state variables involved in fast ...
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### Prove $T_n(x)$ of Chebyshev Polynomial given the recurrence relation [closed]

Using the recursion formula for Chebyshev polynomials, show that $T_n(x)$ can be written as $$T_n(x)=2^{n-1}(x-x_1)(x-x_2)...(x-x_n)$$ where $x_i$ are the $n$ roots of $T_n$ The recurrence relation:...
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### Transfrom a Legendre polynomial from $\int_{-1}^{1}\phi_j(x)\phi_k(x)dx$ into $\int_{a}^{b}\phi_j(t)\phi_k(t)dt$ given $t=\dfrac{1}{2}[(b-a)x+(a+b)]$

The Legendre polynomials satisfy $$\int_{-1}^{1}\phi_j(x)\phi_k(x)dx = \begin{cases} 0 &j\neq k\\\\ \dfrac{2}{2j+1} &j=k \end{cases}$$ Suppose that the best fit problem is given on the ...
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### Choosing good basis functions to approximate a Lipschitz function

Let $D = \left\{0, t_1, t_2, \ldots, t_n\right\} \times [0,1]$ and $$f: D\to [0,1],$$ be a function of time and a one-dimensional space. There is no analytical formula for $f$, but $f(t_i, \cdot)$ ...
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### Newton's method in interpolation

I've seen that in Newton's method for interpolating polynomials, the coefficients can be found algorithmically using (in Python-ish): ...
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### Capacitance in freefem++

I would like to simulate a capacitor in 2d with freefem++. This is the code I used: ...
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### What is the origin of the preasymptotic convergence behaviour in FEM?

When you have fine-scale features (e.g. boundary layers) in the solution, its FEM approximation on coarse meshes converge at strange apparent rates. Looking at Cea's lemma, is this behaviour because ...
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### Is Discrete Exterior Calculus currently a focusing point in numerial computing world or simulation in industry,

I am just wondering if Discrete Exterior Calculus, as a new numerical method , is good at numericall solving problems in elasticity, fluids or other physical/real area.
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### A problem in 1D linear finite element method

When applying Galerkin method, we have two conventions, i.e. multiply the test function $v$ at left/right, $(v,u)/(u,v)$. Both ways won't matter for a simple problem like Poisson's equation, since the ...
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### Significance of p-convergence studies

Consider a method (e.g., FEM) with variable approximation order $p$. Now, we know that the optimal order of convergence is given by $$e = C h^{p+1},$$ where $h$ denotes the mesh size and a constant $C$...
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### Examples of numerical solution of stochastic differential equation(SDE)?

I want to simulate a nonlinear stochastic differential equation $${\rm d}X_t = f(X_t) {\rm d}t + g(X_t){\rm d}B_t$$ where $f,g \in C^{\infty}({\mathbb R}^n ,{\mathbb R})$ and $B_t$ is one-...
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### How do you debug numerical code, what could be source of this oscillatory error?

Quiet a lot of insight can be gained form experience, I was just wondering if anybody has seen something similar to this before. The plot shows the initial condition (green) for the advection-...
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What implementation details need to change if I use a cell average approach rather than a cell total approach for the finite-volume method? For example, consider the conservation law, $$u_t + \... 3answers 4k views ### What's the state of the art in parallel ODE methods? I'm currently looking into parallel methods for ODE integration. There is a lot of new and old literature out there describing a wide range of approaches, but I haven't found any recent surveys or ... 1answer 326 views ### Multivariate numerical integration with a non-uniform grid I want to approximate the integral:$$ I = \int f(\boldsymbol{x})d\boldsymbol{x} $$where \boldsymbol{x} is d-dimensional. I have a set of non-equally spaced points \boldsymbol{x}_1, \dots, \... 1answer 14k views ### How should boundary conditions be applied when using finite-volume method? Following from my previous question I am trying to apply boundary conditions to this non-uniform finite volume mesh, I would like to apply a Robin type boundary condition to the l.h.s. of the domain (... 0answers 237 views ### Block Backward Differentiation Formula (BBDF), on order 4 formula I am trying to implement a program the numerical method to solve ODE called Block BDF as explained in this article: https://waset.org/journals/waset/v38/v38-49.pdf As it is variable step-size, I need ... 1answer 226 views ### Is using Monte Carlo method a good approach for solving Boltzmann equation? I'm trying to solve for electron and hole distribution function using Boltzmann equation with various scattering mechanisms. Since I land up with an integro-differential equation, analytical solution ... 1answer 437 views ### ENO-WENO Schemes: Are ENO-WENO schemes non oscillatory for all kinds (linear/non linear) of problems? Is there an rigorous proof of ENO-WENO schemes being non oscillatory? 3answers 2k views ### How should non-constant coefficients be treated with finite-volume first order upwind scheme? Starting with the advection equation in conservation form.$$ u_t = (a(x)u)_x $$where a(x) is a velocity which depend on space, and u is a concentration of a species which is conserved. ... 4answers 1k views ### Optimal ODE method for fixed number of RHS evaluations In practice, the runtime of numerically solving an IVP$$ \dot{x}(t) = f(t, x(t)) \quad \text{ for } t \in [t_0, t_1]  x(t_0) = x_0 $$is often dominated by the duration of evaluating the right-... 2answers 398 views ### Oscillations in singularly perturbed reaction-diffusion problems with finite elements When FEM-discretizing and solving a reaction-diffusion problem, e.g.,$$ - \varepsilon \Delta u + u = 1 \text{ on } \Omega\\ u = 0 \text{ on } \partial\Omega $$with 0 < \varepsilon \ll 1 (... 1answer 1k views ### Integration of an indefinite integral: matlab precision problem The integral I need to evaluate is:$$ \int_x^{\infty} \frac{t^n}{e^{t} -1} dt $$After some research I found a paper saying, The numerical values of the two integrals [...] are easily calculated ... 1answer 210 views ### Why do I get “estimated error” -1.#IND when doing BICGSTAB linear solver using ILUT perconditioner in eigen I'm using Eigen (a C++ library for numerical linear algebra) to solve a linear equation with the the bi-conjugate gradient BICGSTAB algorithm with Incomplete LU preconditioner. However, the result <... 1answer 797 views ### Solving Coupled ODE eigenvalue problem I've been trying to find some resources that would help me figure out how to numerically solve a coupled system of ODEs which is also an eigenvalue problem. The system is something like:  \tag{1} ... 1answer 180 views ### Local truncation error and transformation of coordinates I am given the advection equation$$ u_t=u_x $$and then the transformation of coordinates$$ x=x(\xi,\theta), \qquad t=\theta $$which leads us to the transformed equation$$ x_{\xi} u_{\theta} - u_{\...
According to Wolfram Alpha and the Sage computer algebra system, the following identity holds:  \cos\left(\arctan\left(\frac{l_1-l_2}{d}\right)\right) = \frac{1}{\sqrt{1 + \frac{(l_1-l_2)^2}{d^2}}} \$...