Nonlinear solvers

Following non-linear solvers are available in OpenGeoSys:

• Newton
• Picard

Newton

The nonlinear solver of “Newton” type is an implementation of the Newton-Raphson method. The basic definition of the non-linear solver with “Newton” follows this template:

<nonlinear_solver>
    <name>basic_newton</name>
    <type>Newton</type>
    <max_iter>10</max_iter>
    <linear_solver>linear_solver</linear_solver>
</nonlinear_solver>

Configuration parameters

• name: Unique identifier for the nonlinear solver.
• type: Must be Newton for this solver type.
• max_iter: Maximum number of nonlinear iterations.
• linear_solver: Reference to a defined linear solver.
• recompute_jacobian: (Optional, default: 1) Frequency of Jacobian recalculation. 1 means recalculate every iteration.
• damping: (Optional, default: 1.0) Damping factor for the Newton update.
• damping_reduction: (Optional) Factor by which the damping is reduced when convergence is slow.
• tikhonov: (Optional) Tikhonov regularisation configuration (see below).

Tikhonov regularisation

Tikhonov regularisation can be used to improve convergence for ill-conditioned systems by adding a value to the diagonal of the Jacobian matrix:

<nonlinear_solver>
    <name>newton_with_tikhonov</name>
    <type>Newton</type>
    <max_iter>100</max_iter>
    <linear_solver>linear_solver</linear_solver>
    <tikhonov>
        <lambda>1e-14</lambda>
        <starting_iteration>11</starting_iteration>
    </tikhonov>
</nonlinear_solver>

The tikhonov element contains:

• lambda: The regularisation parameter (typically 1e-20 to 1e-10).
• starting_iteration: The iteration from which regularisation is applied (default: 0).

Picard

The nonlinear solver of “Picard” type is an implementation of the Picard-Iteration method. The basic definition of the non-linear solver with “Picard” follows this template:

<nonlinear_solver>
    <name>basic_picard</name>
    <type>Picard</type>
    <max_iter>100</max_iter>
    <linear_solver>linear_solver</linear_solver>
</nonlinear_solver>