numerical-methods

Numerical integration is one of the central tools of scientific computing. Whenever a physical, biological, chemical, mechanical, electrical, or control system is modeled by differential equations, the next practical question is usually not only “what is the equation?” but also “which numerical method should be trusted to simulate it?” This question is more subtle than it first appears. A numeric…

New results challenge AI’s promise for solving how fluids swirl—and suggest a more human path forward

Scientific Reports, Published online: 09 June 2026; doi:10.1038/s41598-026-56636-w Numerical simulation of proppant transport in hydraulic fractures with length-dependent variable aperture

I'm trying to get the log ratio between two numbers close to each other ­— say, a = 256.0001 and b = 256.0. Disregarding computational efficiency, which formula would be most numerically accurate? ...

Solving Linear Equations with Gauss Elimination: A Complete Guide with C Code Have you ever wondered how engineering software solves systems of 50 equations with 50 unknowns in milliseconds? The answer at the heart of most linear solvers is the Gauss Elimination Method — one of the most important algorithms in numerical computing. Today I'll break it down from scratch: the concept, the algorithm,…

ALGORITHM: Newton-Raphson Method for Finding Roots INPUT: - Function f(x) - Derivative f'(x) - Initial guess x₀ - Tolerance ε (epsilon) - Maximum iterations N OUTPUT: - Approximate root xₙ - Number of iterations Step 1: START Step 2: Set iteration count i = 0 Step 3: Input initial guess x₀ Step 4: Input tolerance value ε (e.g., 0.00001) Step 5: Input maximum iterations N (e.g., 100) Step 6: DO WH…

Doctoral Thesis Oral Defense - Bailey Mark Miller Gates and Hillman Centers jennsbl Fri, 06/05/2026 - 09:47 6501 & Zoom Monte Carlo Methods for Linear Elliptic Boundary Value Problems BAILEY MARK MILLER This thesis develops the walk on spheres family of Monte Carlo PDE solvers into a practical computational framework for solving linear elliptic boundary value…

Doctoral Thesis Oral Defense - Bailey Mark Miller Gates and Hillman Centers jennsbl Fri, 06/05/2026 - 09:47 6501 & Zoom Monte Carlo Methods for Linear Elliptic Boundary Value Problems BAILEY MARK MILLER This thesis develops the walk on spheres family of Monte Carlo PDE solvers into a practical computational framework for solving linear elliptic boundary value…

Suppose you run across the power series for the exponential function and decide to code it up. Good idea: you’ll probably learn something, though maybe not what you expect. Maybe you decide a tolerance of 10−12 is good enough, and so you sum the terms until the next term to add is below the tolerance. […] The post Naively summing an alternating series first appeared on John D. Cook .

My overall objective is to plot a function that is composed of two individual functions with one function within the other. In a minimum working example I start by making a region and then defining a ...

Why do I want to create a new mapping for unicode-math? I noticed that Unicode has sans-serif arrow keys in U+1F850 - U+1F859 range, but unicode-math and \symsf seem to only superficially include ...

I am interested in numerically evaluating $$g(\mathbf{r}')=\int_{D}\frac{f(\mathbf{r})}{|\mathbf{r}-\mathbf{r}'|^{\gamma}}d^3r$$ with $f(\mathbf{r})$ some continuous (not necessarily analytic) ...

Accurate determination of plane-strain fracture toughness is essential for structural integrity assessments of engineering components, yet conventional ASTM standardized testing methods require large, notched specimens that are impractical in many industrial and research contexts. This thesis presents a finite element method-based approach for estimating the mode I plane-strain fracture toughness…

IntroductionDeveloping offshore deep low-permeability to tight reservoirs faces inconsistent success due to ambiguous reservoir classification and inaccurate production evaluation. This study addresses these issues by analyzing oil wells in the western South China Sea.MethodsThe target reservoir was classified into four types using a flow zone indicator method, and permeability conversion models …

A battle between “slimes” and “zoglins” could be the best way to calculate pi—at least for fans of this megahit game

A couple months ago I wrote about how to compute the sine and cosine of a complex number using only real functions of real variables using the equations You can do something analogous for all the elementary functions, though some of the equations are quite a bit more complicated than the ones above. See the […] The post Building complex functions out of real parts first appeared on John D. Cook .

Scientific Reports, Published online: 22 May 2026; doi:10.1038/s41598-026-53831-7 Parameter-uniform numerical method for a coupled system of singularly perturbed turning point problems with Robin boundary conditions

This paper introduces the pole ratio metric and presents a sphere-based view of symmetric positive-definite matrix rotations on the Riemannian manifold of symmetric positive-definite matrices equipped with the affine-invariant Riemannian metric. The pole ratio quantifies whether data from different users lie on this Riemannian manifold in a way that enables effective transfer learning. The sphere…