Indirect methods approximate the derivatives in ODE/PDEs and then solve those.

Direct methods aim to approximate and solve the ODE/PDE directly.

The Rayleigh-Ritz method is an indirect numerical method.

Finite differences are an example of indirect methods.

All direct methods have stability conditions, which limits their applicability.

... can be computed by applying a displacement of 1 to all nodes simultaneously and then identifying the resulting forces at all nodes.

... depends on the nodal displacements.

... is constant.

... depends only on Young's modulus, the cross-sectional area, and the element's undeformed length.

... is of size 4x4.

... depends only on Young's modulus, the cross-sectional area, the element's undeformed length, and the bar's orientation angle.

... is of size 6x6.

Each node has degrees of freedom in dimensions.

The 2-node bar element is appropriate only for small displacements, because it considers the orientation of the bar to be constant.

Each node has degrees of freedom in dimensions.

The 2-node bar element is appropriate for small and large displacements, because it considers the reorientation of the bar through its displacements.

The stiffness matrix of a 2-node bar element that is oriented vertically (i.e., which points in the -direction) has 6 non-zero entries.

The central difference approximation of the second derivate is second-order accurate.

The forward-Euler approximation of the first derivative is second-order accurate.

All central-difference schemes are first-order accurate.

The fact that the truncation error, e.g. (Δx)ⁿ , must decay as Δx tends towards zero, is referred to as the consistency condition.

According to the Lax-Richtmyer theorem, a finite-difference scheme converges if and only if it is consistent and stable.

Von Neumann analysis yields only a necessary condition for the stability of a finite-difference scheme, so that further analysis is required to arrive at a sufficient condition.

A finite-difference scheme that is consistent must be either conditionally or unconditionally stable.

Explicit schemes tend to be unconditionally stable.

Explicit schemes do not require the solution of a linear system so that each solution step is efficient.

Implicit schemes tend to be unconditionally stable.

Implicit schemes typically admit larger time steps.