Perform spatial and temporal convergence analysis for solution verification — compute observed convergence orders from grid or timestep refinement studies, apply Richardson extrapolation to estimate discretization error, and calculate the Grid Convergence…
Select and apply numerical differentiation schemes for PDE and ODE discretization — generate finite-difference stencils at arbitrary order and accuracy, choose between central, upwind, compact (Pade), and spectral methods, handle boundary stencils, and…
Select and configure linear solvers for Ax=b systems arising in numerical simulations — choose between direct (LU, Cholesky) and iterative (CG, GMRES, BiCGSTAB, MINRES) methods, analyze sparsity patterns and matrix conditioning, recommend preconditioners…
Plan and evaluate mesh generation for numerical simulations — estimate grid resolution from physics scales (interface width, boundary layers, wavelengths), check aspect ratios and skewness against quality thresholds, choose between structured, unstructured,…
Select and configure nonlinear solvers for root-finding f(x)=0, optimization min F(x), and least-squares problems — choose among Newton, Newton-Krylov, quasi-Newton (BFGS, L-BFGS), Broyden, Anderson acceleration, and Levenberg-Marquardt methods, configure…
Select and configure time integration methods for ODE and PDE simulations — choose among explicit Runge-Kutta, BDF, Rosenbrock, and Adams families, set relative and absolute error tolerances, implement adaptive step-size control with P/PI step-size…
Analyze numerical stability for time-dependent PDE simulations — check CFL and Fourier criteria, perform von Neumann stability analysis, detect stiffness, evaluate matrix conditioning, and recommend explicit vs implicit time-stepping schemes. Use when…
Plan and control time-step policies for transient simulations — couple CFL and physics-based stability limits with adaptive stepping, ramp initial transients through sharp gradients or phase changes, schedule output intervals and checkpoint cadence, and plan…
