Concept graph
Every concept is owned by exactly one lesson — enforced at build time. Any lesson needing it links there instead of re-teaching it. Anything required but unowned shows up below as a gap, which is how the to-write queue stays honest.
All concepts
A method is A-stable when it stays bounded on decaying problems for ANY step size.
Estimate the local error each step and resize h to hold it near a tolerance.
Exact derivatives from the computational graph - neither symbolic nor finite-difference.
Evaluate the slope at the destination instead of the origin. Requires solving an equation, and buys unconditional stability.
The compact table of coefficients that fully specifies a Runge-Kutta method.
Subtracting two nearly equal numbers annihilates the leading digits and promotes rounding noise into the answer.
Straddles the point symmetrically so the leading error term cancels; second-order accurate for the same cost.
Four slope samples per step, combined so error terms through third order cancel. The default workhorse.
Step into the imaginary direction to get a derivative with no subtraction, and therefore no cancellation at all.
A property of the PROBLEM: how much the answer moves when the input is perturbed.
Make the solver itself differentiable, so gradients flow through physics and you can optimise through time.
Two methods of different order sharing stage evaluations, so the error estimate is nearly free.
Secular, one-directional loss or gain of a conserved quantity over a long integration.
Reals are stored as a sign, a finite mantissa and an exponent, so almost every number you write down is approximated.
Approximates a derivative from two samples one step apart; first-order accurate.
Step along the tangent you can see right now. The simplest integrator, and the one whose failures teach the most.
The accumulated error at the end of a run. One order lower than the local error, because you take O(1/h) steps.
Mechanics written so that a single function H generates the flow, and phase-space volume is conserved.
Predict with Euler, then correct with the average of both slopes. Second order for two evaluations.
A differential equation plus a starting state, which together determine the whole trajectory.
The error committed by a single step, assuming everything before it was exact.
The gap between 1 and the next representable double - the fundamental resolution of arithmetic.
Integrate many interacting particles to recover thermodynamics from mechanics.
Trade determinism for dimension-independence: error falls as 1/sqrt(N) no matter how many dimensions.
A property of the ALGORITHM: whether it amplifies the errors it makes along the way.
The exponent p in error = O(h^p): how fast the error falls when you refine the step.
Train a network whose loss is the PDE residual, turning solving into optimisation.
Combine two step sizes so the leading error term cancels algebraically, buying an order for free.
Take several trial slopes inside one step and combine them so low-order error terms cancel.
Represent the solution in a Fourier or polynomial basis and get exponential accuracy for smooth problems.
When a system's fastest mode forces a far smaller step than accuracy alone would need.
Forward Euler with the position update using the ALREADY-UPDATED velocity. One reordering, entirely different long-run behaviour.
Methods that exactly preserve the geometric structure of Hamiltonian flow, rather than merely approximating the trajectory.
A symplectic method solves a slightly different problem EXACTLY, which is why its energy error stays bounded.
Shrinking h cuts truncation error but amplifies roundoff, so accuracy bottoms out at a finite optimal step.
Running the integrator backwards returns you exactly to where you started.
Discretise the ACTION rather than the equations of motion, and conservation laws follow automatically.
Second-order, symplectic, time-reversible. The standard integrator for molecular dynamics and N-body.