Iterative Methods for Solving Partial Differential Equations

These flashcards cover key vocabulary and concepts related to iterative methods used for solving partial differential equations, including definitions, methodologies, and important mathematical properties.

Iterative Methods

Techniques used to solve large, sparse matrices in an efficient manner by generating a sequence that converges to the solution.

Finite Difference Methods

Numerical techniques for approximating the solutions to partial differential equations.

Sparse Matrix

A matrix in which most elements are zero, allowing for more efficient storage and computation.

Jacobi Method

An iterative algorithm that updates unknowns by substituting known current values to find next approximations.

Gauss-Seidel Method

An iterative technique that uses the most recent values computed in the solution process to update unknowns.

Successive Overrelaxation (SOR)

A variant of the Gauss-Seidel method that accelerates convergence by forming a weighted average of previous and new values.

Convergence

The property of an iterative method to approach a specific solution as iterations progress.

Spectral Radius

A measurement that can indicate the rate of convergence of an iterative method.

Poisson's Equation

A partial differential equation of great importance in physics and engineering, used in various applications including heat flow and electromagnetism.

Boundary Condition

Constraints necessary in the solution of differential equations that account for the behavior of solutions at the boundaries of the domain.

Diagonal Dominance

A condition in which the magnitude of each diagonal element in a matrix is greater than the sum of the magnitudes of all other elements in that row, ensuring convergence for iterative methods.