Category Simulation & Modeling Projects
Advection-Dominated Problems: Upwind Schemes and SUPG Stabilization
Reading Time: 12 minutesAdvection-dominated partial differential equations are difficult because information moves mainly along a flow direction while physical diffusion provides little smoothing. A numerical method that works well for a diffusion-dominated problem may produce oscillations, excessive smearing, or nonphysical values when advection becomes stronger. The correct response is not simply to select the most diffusive scheme available. […]
GPU Kernel Programming for Custom Physics Simulation: Writing High-Performance Custom Kernels in Python
Reading Time: 11 minutesLearn to write custom GPU kernels for physics simulation with NVIDIA Warp, CUDA Python, and HIP/ROCm. Covers performance patterns, differentiable physics, and production deployments.
Implicit vs Explicit Methods: Stability, Accuracy, and When to Use Each
Reading Time: 10 minutesChoosing a time-integration method is one of the most important decisions in a scientific simulation. The method determines how the numerical solution moves from one time level to the next, how small the time step must be, how much each step costs, and whether fast physical processes are resolved or suppressed. The common distinction between […]
Benchmark Suites for Scientific Solvers: SciML, DOE Sparse Solvers, and ASU Mittelmann
Reading Time: 8 minutesBenchmarking is essential when researchers need to compare numerical solvers, machine learning models, sparse linear algebra libraries, or optimization software. A solver that performs well on one equation, matrix, tolerance, or hardware platform may behave very differently under another set of conditions. A useful benchmark should provide repeatable problem definitions, consistent accuracy measurements, controlled computing […]
Spectral Methods for PDE Simulation: Theory, Implementation, and When to Use Them
Reading Time: 10 minutesSpectral methods are high-order numerical techniques for solving partial differential equations. Unlike finite difference methods, which approximate derivatives through local stencils, spectral methods represent the solution with basis functions that extend across the entire computational domain. Common choices include Fourier modes for periodic problems and Chebyshev or Legendre polynomials for bounded, non-periodic domains. When the […]
Uncertainty Quantification and Sensitivity Analysis in Scientific Simulation
Reading Time: 11 minutesLearn polynomial chaos expansion, Sobol sensitivity indices, surrogate UQ methods, and the Python ecosystem. A practical guide for computational researchers.
Stiff Problems and Stiff Solvers: When Explicit Methods Fail
Reading Time: 11 minutesWhen explicit simulations blow up, stiffness is usually the culprit. Learn why explicit methods fail, how to diagnose stiffness, and which solvers actually work.
Inverse Problems and Parameter Estimation for PDE Models
Reading Time: 13 minutesLearn how to estimate unknown parameters from PDE simulations using adjoint methods, Bayesian inversion, and Python packages like TRIPs-Py and dolfin-adjoint.
Operator Splitting, Strang Splitting, and IMEX Schemes for PDE Solvers
Reading Time: 9 minutesTime-dependent partial differential equations often combine several physical processes. A transport model may include advection, diffusion, chemical reactions, external sources, and nonlinear feedback. Each part can have different mathematical properties and may require a different numerical treatment. Advection is commonly handled with methods designed for wave-like transport. Diffusion often creates stiffness and benefits from implicit […]
Comparing Continuum vs. Discrete Modeling Approaches
Reading Time: 6 minutesScientific models simplify complex systems so researchers and engineers can study behavior, test assumptions, and predict outcomes. One of the most important modeling choices is whether to describe a system as a continuous medium or as a set of separate components. A continuum model represents properties such as pressure, temperature, density, velocity, stress, or concentration […]