Andrés Ávila (Universidad de La Frontera, Chile): Global superlinear linearization schemes based on adaptive strategies for solving Richards’ equation
Zoom Link:
https://uni-kassel.zoom-x.de/j/96217091997?pwd=RVRaVVBRclFJYU9jczNZSWF3SXI2QT09
Meeting ID: 962 1709 1997
Passcode: cauchy
Abstract:
The Richards’ equation is a nonlinear degenerate parabolic differential equation, whose numerical solutions depend on the linearization methods used to deal with the degeneracy. Those methods have two main properties: convergence (global v.s. local) and order (linear v.s. quadratic). Among the main methods, Newton’s Method, the modified Picard method, and the L-scheme have one good property but not the other. Mixed schemes get the best of both properties, starting with a global linear method and following with a quadratic local scheme without a clear rule to switch from a global method to a local method.
In this work, we use two different approaches to define new global superlinear and quadratic schemes. First, we use an error-correction convex combination of classical linearization methods, a global linear method and a quadratic local method by selecting the parameter λnk via an error-correction approach to get fixed point convergent sequences. Second, we use a parameter τ to adapt the time step in the general Newton-Raphson method applying to three classical linearizations and the new three error-correction linearizations. Finally, we consider a combination of the L-scheme and the τ-adaptive Newton’s Method, mixing both of both methods. We first built an error-correction type-Secant scheme (ECtS) without derivatives to get a superlinear global scheme. Next, we build the convex combination of the L-scheme with three global schemes: the type-Secant scheme (ECLtS), the modified Picard scheme (ECLP), and the Newton’s scheme (ECLN) to obtain global superlinear convergent schemes. For the second approach, we first apply the τ-adaptation to the classical methods (τ-Newton, τ-L-scheme, and τ-modified Picard), next we apply to the error-correction schemes (τ-AtS, τ-ALtS, τ-ALP, τ-ALN) and finally a new combination of the L-scheme with the τ-Newton adaptive method (τ-LAN).
We test the twelve new schemes with five examples given in the literature showing that they are robust and fast, including cases when the Newton’s scheme does not converge. Moreover, we include an example which uses the Gardner exponential nonlinearities, showing that L- and L2-schemes are too slow as linearization techniques. Some new schemes show high performance in different examples. The τ-LAN scheme has advantages, using fewer iterations in most examples.
References
[1] M. Amreim and T. P. Wihler. An Adaptive Newton-Method based on a dynamical systems approach. Communications in Nonlinear Science and Numerical Simulation, 19:2958–2973, 2014.
[2] G.Albuja and A. I. Ávila. A family of new globally convergent linearization schemes for solving Richards’ equation. Applied Numerical Mathematics, 159:281–296, 2021.
[3] F. Radu, I.S. Pop, and P. Knabner On the convergence of the Newton method for the mixed finite element discretization of a class of degenerate parabolic equation. Numerical Mathematics and Advanced Applications, 42:1194–1200, 2006.
Mit freundlichen Grüßen,
Prof. Dr. Andreas Meister