Pseudorelaxation

Pseudorelaxation is a numerical technique used to accelerate the convergence of iterative methods for solving systems of linear equations, particularly in contexts like computational fluid dynamics or finite element analysis. It is a variation of the Gauss-Seidel method, which is itself an iterative approach to solving linear systems.

The core idea behind pseudorelaxation is to introduce a parameter, often denoted by omega (ω), which modifies the update step for each variable. In a standard iterative method like Jacobi or Gauss-Seidel, the new value of a variable is calculated directly from the current values of other variables. Pseudorelaxation, however, takes a weighted average of the new value computed by the underlying iterative method and the previous value of the variable.

The formula for pseudorelaxation can be expressed as:

x_i^(k+1) = (1 - ω) * x_i^(k) + ω * x_i_new

Here, x_i^(k+1) is the updated value of the i-th variable at iteration k+1, x_i^(k) is its value at iteration k, and x_i_new is the value computed by the underlying iterative method (e.g., Gauss-Seidel) at iteration k+1. The parameter ω is called the relaxation factor.

If ω = 1, pseudorelaxation reduces to the standard Gauss-Seidel method. If 0 < ω < 1, it is called under-relaxation, which can help stabilize iterative processes that might otherwise diverge. If 1 < ω < 2, it is called over-relaxation, which can significantly speed up convergence for certain types of problems, particularly those arising from discretized partial differential equations. The optimal choice of ω depends heavily on the specific properties of the linear system being solved. Choosing an inappropriate ω can lead to slower convergence or even divergence.

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