Pseudo-Transient Method

The Pseudo-Transient Method (PT method), is an iterative method which is:

matrix-free
builds on a fixed-point iteration
- update of each grid point is local, does not require global reductions
enables easy-to-develop multi-physics coupling due to its conciseness
similarity between mathematical and discretised code notation

History

1911 - Pioneering work by Richardson

Richardson proposed an iterative solution approach to PDEs related to dam-engineering calculations. Early developed iterative algorithms are well-suited for early low-memory computers but lack in efficient convergence rates.

1950 - First present of the PT method in the literature (Frankel)

The idea of accelerating the convergence by increasing the order of PDE dates back to the work by Frankel (1950). Frankel noted the analogy between the iteration process and transient physics. And the accelarated method was called the second-order Richardson method.

Introduced as an extension of the Richardson and Liebmann methods, with dependency on the previous iterations added.

1965 - The PT method originated as a dynamic-relaxation method (Oter)

The PT method was applied for calculating the stresses and displacements in concrete pressure vessels.

1972 - Enhanced convergence rates of the PT methods showed (Young)

The PT method was firstly performed on par.

1976 - First introduction in geosciences (Cundall)

The PT method was introduced by Cundall as the Fast Lagranngian Analysis of Continua (FLAC) algorithm

1993, 1994 - Applications of the FLAC method (Poliakov et al.)

The FLAC method was successfully applied to simulate the Rayleigh–Taylor instability in visco-elastic flow (Poliakov et al. 1993), and the formation of shear bands in rocks (Poliakov et al. 1994).

1993 - Application in buckling (Ramesh and Krishnamoorthy)

1999 - Application in form-finding (Barnes)

2009 - Application in failure (Kilic and Madenci)

2011 - FEM community still referenced it as the DR-method (Rezauee-Pajand)

2020 - Review on the accurate estimate of extremal eigenvalues for the Chebyshev's semi-iterative methods (Saad)

NOTE: second-order or extrapolated methods are also termed semi-iterative.

2022 - Accelerated pseudo-transient method (Räss et al.)

Assessing the robustness and scalability of the accelerated pseudo-transient method Räss et al. (2022)

Classification of the PT Method

The PT method can be classified differently as the first-order PT method and the accelerated PT method.

The first order PT method performs the pseudo-time stepping based on a first-order scheme by introducing the pseudo-transient term in the form of $\frac{\partial}{\partial \tau}$. A physical property A is updated at each PT iteration, using the current values of pseudo-time steps and residuals.

Accelerated PT Method

The accelerated PT method is based on a second-order scheme, where a pseudo-transient term in the form of $\alpha \frac{\partial^2}{\partial \tau^2} + \frac{\partial}{\partial \tau}$ is introduced. The name of the method comes from the fact that it can significantly enhance the convergence rates of the algorithm when selecting the appropriate damping parameter α.

Followingly we abstract some important aspects reported in the paper "Assessing the robustness and scalability of the accelerated pseudo-transient method Räss et al. (2022), in which the accelerated PT method was introduced.

It has the following advantages:

Ensures the iteration count to scale linearly with numerical resolution increase

Application

The method is applicable to:

Strongly nonlinear problems
- shear-banding in a visco-elasto-plastic medium

Finding solution of stationary problems
Finding solution of problems with transient terms
- involve both physical time $t$ and pseudo-time $\tau$
- also called "dual-time method" or "dual time stepping" (Mandal et al 2011)

Derivation

A physically motivated derivation is well-presented . To understand how powerful the PT method is, we cited here a small paragraph from the paper:

"The PT methods build on a physical description of a process. It therefore becomes possible to model strongly nonlinear processes and achieve convergence starting from nearly arbitrary initial conditions."

The accelerated PT method for elliptic equations is mathematically equivalent to the second-order Richardson rule.