Scientific Computing · Cluster
Differential Equations
Discretization and stability.
- 01 Local Truncation Error in the Explicit Euler Method Where the O(h^2) per-step error in forward Euler comes from, why the global error drops to O(h), and how roundoff limits the step size from below.
- 02 Global Error Accumulation and the CFL Condition How local truncation errors combine across steps, why the von Neumann analysis of explicit advection produces the Courant number bound, and what propagation velocity has to do with it.
- 03 Stiffness: When Explicit Steps Collapse What makes an initial value problem stiff, why an explicit method is forced to steps far below what accuracy requires, and how the Jacobian eigenvalues set the forced step.
- 04 Butcher Tableaux and the Order Conditions of Runge-Kutta Methods How a Runge-Kutta scheme is encoded in a tableau, which algebraic relations among its coefficients produce each order of accuracy, and why the condition count grows so fast.
- 05 A-Stability and the Dahlquist Barrier Why the second Dahlquist barrier limits linear multistep methods to order two when they must damp every decaying mode, and how the equivalent restrictions apply to Runge-Kutta schemes.
- 06 Adaptive Step Size Control with Embedded Runge-Kutta Pairs How an embedded pair produces an error estimate from the same stages, why the controller takes the square root of the tolerance ratio, and how rejected steps and saturation are handled.
- 07 The Parabolic Step Limit in Explicit Diffusion Schemes Why an explicit update of the heat equation requires a step proportional to the square of the mesh spacing, what the amplification analysis shows, and how implicit treatment removes the restriction.
- 08 Newton Iterations for Implicit Spatial Operators How an implicit step on a nonlinear PDE becomes a Newton solve on a sparse system, what convergence the iteration has, and why the Jacobian is assembled from the stencil.
- 09 Symplectic Integrators and the Shape of Energy Drift Why a symplectic method does not conserve energy exactly but keeps it bounded, how the modified Hamiltonian explains the bound, and when a non-symplectic method can be better.
- 10 Limiters, the TVD Condition, and Shock Capture Why a second-order scheme oscillates at a discontinuity, how the total variation diminishing condition forbids new extrema, and what a slope limiter must satisfy to preserve both properties.