📘 How do physicists compute with machines?

What you’ll learn

1. Finite-Difference Time-Domain for Maxwell EquationsDerive dispersion relations and stability limits for FDTD schemes applied to electromagnetic wave propagation.The chapter quantifies numerical dispersion and anisotropy in the Yee lattice. It demonstrates how a fourth-order time integrator reduces phase error by two orders of magnitude at fixed cell size. Stability diagrams and concrete CFL numbers for dispersive media are derived from the amplification matrix.
2. Spectral Methods for the Schrödinger EquationImplement Fourier and Chebyshev spectral discretizations and control aliasing errors in quantum eigenvalue problems.The lesson derives the spectral differentiation matrices for periodic and bounded domains. It shows how dealiasing via the 3/2 rule restores exponential convergence for the nonlinear Hartree term. Benchmark calculations on the helium atom illustrate residual errors below chemical accuracy.
3. Metropolis Monte Carlo with Cluster AlgorithmsDesign and analyze cluster Monte Carlo algorithms for continuous-spin and lattice-gauge models.The chapter constructs the bond-activation probabilities for O(n) and SU(2) models. It proves detailed balance for the Wolff single-cluster variant and measures integrated autocorrelation times on published 128^3 lattices. Finite-size scaling collapses are performed to extract critical exponents.
4. Symplectic Integrators in Molecular DynamicsConstruct and benchmark symplectic integrators for Hamiltonian systems with multiple timescales.The lesson derives the Baker-Campbell-Hausdorff expansion for splitting methods. It quantifies long-term energy drift versus timestep for Verlet, velocity Verlet, and r-RESPA on a solvated protein. Backward error analysis supplies rigorous bounds on shadow-Hamiltonian fluctuations.
5. MPI Domain Decomposition for N-Body GravityImplement and tune distributed-memory tree codes for self-gravitating systems.The chapter details the orthogonal recursive bisection and Hilbert-curve load-balancing strategies. Communication volume is measured for both particle migration and multipole exchange. Strong-scaling curves on published 10^9-particle datasets illustrate the crossover to communication-bound regimes.
6. GPU Kernels for Lattice QCDOptimize CUDA kernels for Dirac operators and measure sustained performance on modern GPUs.The lesson walks through memory-layout choices, warp-level reductions, and tensor-core utilization for staggered and Wilson fermions. Roofline models are constructed from measured arithmetic intensity. Published performance numbers from the MILC and Chroma codes are reproduced and extended.
7. Numerical Stability in Long-Term N-Body IntegrationsAnalyze and mitigate round-off and truncation errors in symplectic planetary integrators.The chapter derives the modified-equation error terms for the Wisdom-Holman map. It quantifies floating-point accumulation using compensated summation and Kahan-Neumaier techniques. Published 10 Myr integrations of the Jovian planets illustrate the transition from truncation- to round-off-dominated regimes.
8. Machine-Learned Interatomic PotentialsTrain and validate equivariant neural-network potentials for molecular-dynamics production runs.The lesson constructs rotationally equivariant descriptors and loss functions that enforce energy conservation. It reports force, energy, and virial errors on published silicon and water datasets. Uncertainty quantification via query-by-committee drives an active-learning workflow that reaches 1 ns/day throughput.
9. Adaptive Mesh Refinement in Numerical RelativityImplement block-structured AMR and constraint-preserving boundary conditions for the Einstein equations.The chapter derives the prolongation and restriction operators that maintain the BSSN constraints to machine precision. It measures convergence order on the head-on collision benchmark and quantifies the overhead of regridding relative to the evolution timestep.
10. Stochastic Differential Equations for Brownian DynamicsDerive and implement structure-preserving integrators for multiplicative SDEs in colloidal and polymer systems.The lesson constructs the Fokker-Planck operators for both additive and multiplicative noise. It proves that the BAOAB scheme exactly preserves the equilibrium distribution for quadratic potentials. Published mean-square-displacement curves for DNA in nanochannels validate the method.
11. Verification Benchmarks from the NIST AMR LibraryApply rigorous verification procedures using manufactured solutions and grid-convergence studies.The chapter reproduces the MMS source terms for the compressible Navier-Stokes equations. Richardson extrapolation and the grid-convergence index are evaluated on the published NIST datasets. Discrepancies larger than 2 percent are traced to limiter activation near discontinuities.
12. Quantum Algorithms for the Hubbard ModelMap fermionic Hamiltonians to qubit operators and estimate resource requirements for fault-tolerant phase estimation.The lesson constructs the Jordan-Wigner and Bravyi-Kitaev mappings and counts T-gate resources for the 2D Hubbard model. It reports VQE energies on superconducting hardware and compares them with exact diagonalization. Extrapolated logical-qubit counts for 10 percent doping are derived from surface-code overhead models.
13. Exascale I/O and In-Situ AnalysisDesign scalable I/O and in-situ workflows that keep pace with exascale arithmetic throughput.The chapter evaluates collective I/O, compression ratios, and staging strategies on published 10^5-core runs. It quantifies the trade-off between in-situ reduction and post-hoc flexibility using the SENSEI interface. Published exascale projections for fusion and astrophysics codes are reproduced.
14. Uncertainty Quantification in Multiscale Plasma ModelsConstruct surrogate models and perform global sensitivity analysis for kinetic plasma simulations.The lesson derives the stochastic Galerkin and collocation formulations for the Vlasov equation. Sobol indices are computed from the published QUESO library on a 5-species argon discharge. Computational cost is reduced from 10^5 to 200 solver evaluations while preserving 95 percent of the output variance.

Questions this course answers

Grounded in trusted sources

• OpenStax
• NIST
• NASA
• NIST Digital Library of Mathematical Functions / numerical recipes context
• NASA — scientific computing / modeling resources
• OpenStax University Physics — computational examples

Every Wunder lesson is built from real, reputable sources — never invented.

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