Abstract
The most commonly used method by the geodynamical community for solving non-linear equations is the Picard fixed-point iteration. However, the Newton method has recently gained interest within our community because it theoretically yields quadratic convergence close to the solution as opposed to the global linear convergence of the Picard iteration. In mantle dynamics, a blend of pressure and strain-rate dependent visco-plastic rheologies is often used. While for power-law rheologies the Jacobian is theoretically guaranteed to be Symmetric Positive Definite (SPD), for more complex rheologies, especially in combination with compressible models, the Jacobian may become non-SPD. In practice though, even power-law rheologies may yield a non-SPD Jacobian in some extreme cases. Here we present a new method for efficiently enforce the Jacobian to be SPD, necessary for our current highly efficient Stokes solvers, with a minimum loss in convergence rate. Furthermore, we show results for both incompressible and compressible models. The figure below shows the viscosity (top) and the factor used to force the Jacobian to be positive definite, the SPD factor (bottom), for the toughest case (η1 = 5 × 1025 and U0 = 12.5) of the Spiegelman et al. [1] benchmark. A value of one in the SPD factor figure means that the Jacobian is not changed, while a lesser value scales the derivative in the assembly of the Jacobian down, with a zero value resulting in a normal Picard iteration in that location. As the figure shows, a significant scaling is needed to keep the Jacobian positive definite in this case.
Original language | English |
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Publication status | Published - 30 Aug 2017 |
Event | Nethermod: XV international workshop on modelling of mantle and lithosphere dynamics - Hotel Postillion, Putten, Netherlands Duration: 27 Aug 2017 → 31 Aug 2017 https://nethermod.sites.uu.nl/ |
Conference
Conference | Nethermod |
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Country/Territory | Netherlands |
City | Putten |
Period | 27/08/17 → 31/08/17 |
Internet address |