Category Simulation & Modeling Projects
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.
Nonlinear Solver Methods: Newton, Broyden, and Quasi-Newton for PDEs
Reading Time: 10 minutesLearn how to choose between Newton-Raphson, Broyden's quasi-Newton, and Jacobian-free Newton-Krylov methods for solving nonlinear PDE systems. Includes practical Python examples and decision frameworks.
Model Calibration Techniques for Computational Materials Science
Reading Time: 10 minutesLearn Bayesian MCMC, adjoint optimization, surrogate modeling, and practical workflows for calibrating computational materials models.
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.
Error Estimation and Verification Methods: From Truncation Error to Code Verification
Reading Time: 8 minutesKey Takeaways Truncation error is the foundation of all code verification — understanding Taylor series derivation explains why MMS works and why convergence rates match theoretical predictions. The Method of Manufactured Solutions (MMS) is now automated — frameworks like MOOSE and FEniCSx use SymPy to derive source terms symbolically, eliminating the hand-calculation errors that plagued […]
Data Assimilation Methods: A Guide for Computational Scientists
Reading Time: 12 minutesLearn data assimilation methods—EnKF, 3D-Var, 4D-Var, particle filters, and hybrid approaches. A practical guide for computational scientists.
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 […]
When to Use FEM, FVM, or FDM: A Practical Comparison for Beginners
Reading Time: 7 minutesKey Takeaways FEM, or the Finite Element Method, is strong for complex shapes and structural problems where geometric flexibility matters. FVM, or the Finite Volume Method, is a natural choice for fluid dynamics and transport problems because it supports local conservation of mass, energy, and momentum. FDM, or the Finite Difference Method, is the simplest […]
Mesh Quality and Convergence Studies: A Hands-On Guide for Scientific Simulations
Reading Time: 9 minutesTL;DR Before running any scientific simulation, verify that the mesh passes geometric quality checks and then prove that the results do not change significantly with further refinement. A mesh convergence study using at least three grid levels can quantify discretization error through Richardson extrapolation and the Grid Convergence Index. This guide explains the key metrics, […]