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Questions tagged [numerics]

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32 views

Floquet theory for periodic delay differential equations: current numerical routines

I would like to determine the stability of a system of periodic delay differential equations (a seasonal host-parasite model). I've tried to implement the method described in Lemma 2.5 in this paper: ...
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0answers
24 views

Discrete model of cell - cell communication

I am trying to understand how cell to cell communication is studied using a discrete modelling framework. Could someone please suggest suitable references or libraries which already have ...
2
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1answer
45 views

Finite element modelling of thermal expansion of 3D solid bodies

I want to solve the thermal expansion of solid by using FEM approach. When I developed the model based on the the principle the minimum potential energy, the solutions for thermal expansion are not ...
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1answer
161 views

Error using scipy.integrate.solve_ivp: index error: the index 0 is out of bounds for axis 0 with size 0

I am recently working on the code based on the stick-slip phenomenon in Python. It's the stick-slip oscillator (chapter 3.4) in https://www.sciencedirect.com/science/article/pii/S0888327020301205. And ...
1
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1answer
111 views

SIPG method for $-\nabla \cdot (\nu \nabla u)=f$

Consider the diffusion equation with a coefficient $\nu$: $$-\nabla \cdot (\nu \nabla u)=f$$ with Dirichlet boundary conditions $u = g_D$ in $\partial \Omega$. If the coefficient would be constant, ...
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1answer
340 views

Why is the central difference method dispersing my solution?

I am solving numerically the ODE $\ddot x(t)=-c\dot x(t) -\sin(x(t))+F\cdot \cos(\omega t), \;\dot x(0)=x(0)=0$ for $t\in [0,20\pi]$ on an $N=2000$ dimensional grid. I am working on Python, and I ...
2
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2answers
124 views

How is FEM used in structural engineering?

I have learned about the finite element method (FEM) as a method for solving boundary problems given by a PDE. The way I learned it is to approximate the solution by a linear combination of test ...
3
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1answer
72 views

"Optimal" domain partitioning in domain decomposition algorithms

When solving a PDE numerically by domain decomposition methods, what is the "optimal way" to split the domain? Are there any results stating that a particular partition of the domain yields &...
3
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1answer
155 views

How to include negative number in the log-sum-exp?

I want to know summation of some small numbers, such as {e^-1000, -e^1001, e^1002...} If all numbers are positive, I can use log-sum-exp algorithm. But unfortunately, negative numbers are also ...
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0answers
31 views

Weird behavior in for solving TISE in harmonic oscillator potential using the shooting method

I was solving the time independent Schrödinger equation using the shooting method for harmonic oscillator potential. This is the code that I wrote for that with the results (code is written in julia): ...
9
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1answer
438 views

How to solve a second order differential equation (diffusion) with boundary conditions using Python

I am having trouble implementing a model from a publication. Huang, K-L.; Holsen, T.M.; Selman, J.R. Ind. Eng. Chem. Res. 2003, 42, 15, 3620–3625 scihub link: https://sci-hub.se/10.1021/ie030109q I ...
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3answers
5k views

How to get ODE solution at specified time points?

The code below basically illustrates my problem. It is a test code for a pendulum. I solve it using a method suggested on https://stackoverflow.com/questions/12926393/using-adaptive-step-sizes-with-...
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2answers
232 views

Benchmark problems for eigenvalue reordering algorithms sought

Every real matrix $A$ can be reduce to real Schur form $T = U^T A U$ using an orthogonal similiary transform $U$. Here the matrix $T$ is quasi-triangular form with 1 by 1 or 2 by 2 blocks on the main ...
18
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1answer
2k views

How to Run MPI-3.0 in shared memory mode like OpenMP

I am parallelizing code to numerically solve a 5 Dimensional population balance model. Currently I have a very good MPICH2 parallelized code in FORTRAN but as we increase parameter values the arrays ...
6
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1answer
154 views

Compile-time error control vs. interval arithmetic?

I use interval arithmetic for reliable computing. Now, a procedure coded in a good implementation of interval arithmetic takes perhaps about eight times as much as the same procedure carried out ...
7
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2answers
1k views

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-...
2
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1answer
736 views

Error in Simpson's 3/8 rule is higher than that of Simpson's 1/3 rule

For a given function $f(x)$, I have tried to find its numerical integral using Simpson's 1/3 and Simpson's 3/8 rules. I then compare the solution from the numerical quadratures to the analytical ...
3
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0answers
55 views

Typo in a-priori error estimate in a Discontinuous Galerkin paper

I'm looking at this famous paper which is available in the link below: Franco Brezzi, LD Marini, Endre Süli, Discontinuous Galerkin methods for first-order hyperbolic problems, Mathematical Models ...
5
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1answer
174 views

Discrete divergence free functions

I'm studying the weak formulation of NS equations. During the analysis, the book I'm using (Quarteroni-Valli, page 301-302), defined $$Z_h=\{v_h \in V_h: (\operatorname{div}(v_h),q_h)=0 \quad \forall ...
0
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1answer
107 views

interface value on the error equation

https://www.jstor.org/stable/pdf/2157482.pdf, here I have a problem in last equation of (2.6) in section (2.1). When they are considering error equation on the interface $\Gamma$ they get $e_v^{(n)} = ...
2
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1answer
958 views

Local truncation error of Dufort Frankel Scheme

The scheme is given by $$\frac{v_m^{n+1}-v_m^{n-1}}{2k} + b\frac{v_m^{n+1}+v_m^{n-1}-v_{m-1}^n-v_{m+1}^n}{h^2} = 0$$ where $v_m^n$ is the numerical solution at the $m^\text{th}$ spatial coordinate ...
18
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4answers
2k views

Why do we usually not want the eigenvalues of non-symmetric matrices?

I came across this line in a class note I am reading where it discusses finding eigenvalues of matrices. In reality we don't go all the way with Arnoldi. We stop at a decent value of 𝑘. Then the 𝑘 ...
3
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0answers
79 views

Numerical Soultion to Background equations of cosmology

I am trying to solve the background equations of cosmology numerically using Runge-Kutta Dormand Prince method with simplified assumption $8\pi G=1$ and $c=1$. The equations are $$\ddot a = - \frac{1}{...
1
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0answers
87 views

Efficient multidimensional numerical integration in R and C++

I'm trying to perform a 4-dimensional numerical integration in R using a function I wrote in C++ code which is then sourced in <...
1
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2answers
122 views

Imposing pressure variation instead of Dirichlet boundary conditions on Finite Element Method

I always see Finite Element codes solving PDE with Dirichlet or Neumann boundary conditions. But, I have a problem now consisting of a straight cylinder with a circular base (a simple 3D tube), with ...
4
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4answers
234 views

Learning computational science through guided discovery

I am currently trying to get through Pattern Classification by Duda et al (for a course). However, the book seems too dense for me. Pattern recognition seems like a topic that could be better learned ...
3
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1answer
2k views

Applying Neumann boundaries to Crank-Nicolson solution in python

Consider the heat equation $$u_t = \kappa u_{xx}$$ with boundary conditions of $$u(x,0)=0\\ u(0,t)=100\\ u(l,t)=0$$ Numerical analysis by pyton can be done with ...
5
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0answers
144 views

About the condition $\ker(B_h) \subset \ker(B)$ in mixed finite elements formulation

I'm studying mixed finite elements. The problem is a classical saddle-point one: we seek for $(u,p)$ in $V \times Q$: $$A u + B^t p = f$$ $$Bu = g$$ where $A: V \rightarrow V', B:V \rightarrow Q'$ ...
1
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1answer
313 views

Robin Boundary Condition with Implicit Upwind - Finite Difference Method for 2D Convection-Diffusion Equation

I am trying to solve a problem with 2D Convection-Diffusion equation with U = Concentration ($mg/m^{2}$) using Implicit Upwind Finite Difference Method like this $$ \frac{\partial U}{\partial t} + ...
1
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2answers
165 views

Sum of norms over elements is not equal to norm over the whole $\Omega$

In my finite element notes, after the proof of the global estimate for the interpolation error, assuming a regular triangulation with triangles $T_m$: $$\sum_m|v - \Pi_h^r v|_{s,p,T_m} \leq \sigma^{-s}...
1
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1answer
88 views

Classical global estimate for $H^1$ error

I'm having lots of troubles in understanding the proof the estimation of the classical $H^1$ error using finite elements of degree $r$. $$||u-u_h||_{H^1(\Omega)} \leq \frac{M}{\alpha} C h^r |u|_{H^{r+...
7
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0answers
143 views

Is there a graphical interpretation or explanation of automatic differentiation compared to numerical differentiation

I have been looking at automatic differentiation for solving differential equations lately. I understand the basic ideas of using Dual numbers and such for finding derivatives, etc. However, I feel ...
4
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2answers
162 views

What is the difference between $u_h$ and $I_h(u)$ in finite element literature?

In finite element books, we have estimates for $$||u-u_h||$$ and also estimates for $$||u - I_h(u)||$$ where $I_h(u): V \mapsto V_h$ projects a function from the infinite dimensional space to the ...
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0answers
50 views

How to numerically solve PDE that governs the free vortex wake model?

Crossposted at Math SE I am reading a paper on the free vortex wake model for a helicopter rotor blade, which is described by the following PDE: $$\frac{\partial \vec{r}}{\partial \psi} (\psi, \zeta) ...
2
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1answer
630 views

1-D turbulent energy spectra in homogeneous direction (non-isotropic)

I am trying to compute the one-dimensional energy spectra for my channel-flow simulation. I have already written a post-processing script to achieve this; however, I need to validate my code before ...
2
votes
1answer
110 views

How to couple the vibro-acoustic equations by Mortar method for non-matching meshes?

Assume we have two domains $\Omega_a$ a acoustic domain with boundary $\Gamma_a$ and $\Omega_s$ a domain of a solid body with boundary $\Gamma_s$. $\Omega_a$ and $\Omega_s$ have the common interface $\...
0
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1answer
65 views

How do you find the Jacobian matrix of a coordinate transformation given only dynamical data points?

Suppose you have a record of coordinates X(t) for every s units of time from 0 to time T. Suppose in addition you have more data Q(T) that is supposed to be output from some complex numerical ...
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0answers
48 views

How to prove the stability of the interpolation?

From the book (Vidar Thomee, Galerkin finite element methods for parabolic problems), there holds (see Lemma 13.3) \begin{align}\label{eq} \|\nabla I_h u\|_{L_{\infty}}\leq C\|\nabla u\|_{L_{\infty}},\...
5
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0answers
362 views

Fast integration scheme for path integral of Gaussian over a cubic curve

I need to numerically compute an integral of the following form: $$\int_0^1 \frac{1}{2\pi\sigma^2}\exp\left(-\frac{\|(q_0t^3 + q_1t^2 + q_2t + q_3) - a\|^2}{2\sigma^2}\right)\|3q_0t^2 + 2q_1t + q_2\|\,...
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0answers
58 views

Ritz projection error estimate for bilinear Lagrange element?

For second order elliptic problem \begin{align} -\Delta u=f,\quad in ~~\Omega\\ u=0,\quad on~~ \partial\Omega, \end{align} we have for the Ritz projection for $P_1$ conforming element \begin{align} \|...
2
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0answers
43 views

Does the time-dependent 1D advection-diffusion with point sources have an analytical solution?

I am looking for the analytical solution of 1-dimensional advection-diffusion equation with several point sources, Q, along the axial length of a cylinder through which the fluid flow occurs. Neumann ...
2
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2answers
272 views

How to implement point source or volume source in finite element implementations

I'm trying to do a simple implementation to study the advection-diffusion-reaction dynamics in a straight pipe. I have points positioned along the length of the pipe (blue dots in the image above). I ...
2
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1answer
194 views

Another way to evaluate the gravitational force from a uniform cube?

Appendix A of Liu, Baoyin, and Ma (2011) Equilibria, periodic orbits around equilibria, and heteroclinic connections in the gravity field of a rotating homogeneous cube shows an analytic expression ...
5
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4answers
899 views

How important is learning hardware/architecture for scientific computing?

Apologies if this is a bit of a soft, unclear, or opinion-based question. I'm a relatively new PhD student in a (computational) quantum chemistry group. My group develops and maintains a few software ...
0
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1answer
33 views

Numerical algorithm to calculate stream of discounted cash flows

Is it possible to calculate the flow of cash in each period given a certain fixed net present value with a numerical algorithm? I would like to be able to apply a certain weighting that can be applied ...
0
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1answer
122 views

Interpreting multivariable root-finding results from Matlab's fsolve algorithm

Edit: So I was able to get the same value of r that's given, when coding up the sum of squares of function values directly in the script file, rather than on the Command Window. So, maybe there's a ...
1
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0answers
60 views

For a hyperbolic PDE, is there any proof that the BDF2 method is stable for integrating them?

I would like to ask a question on the stability of BDF2 applied to hyperbolic PDEs. Say I have a hyperbolic equation as $\frac{\partial c}{\partial t} + {\bf U} \cdot {\bf \nabla}c=0$. This system is ...
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0answers
37 views

Expression for numerical amplification factor for Euler time integration and CD2 scheme for one degree hyperbolic function

This is the expression to be derived but I am not getting the exact expression as given in the image. Is the given expression given for the implicit Euler method? Here $N_c = c (\delta t)/h$ and $\tan(...
4
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0answers
79 views

Unstable Algorithms which become stable when hardware provides Kulisch exact dot product instruction

In John Gustaffson's book The End of Error, he discusses Ulrich Kulisch's exact dot product, which (in double precision) requires a 2100 bit fixed point register which rounds only once after the ...

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