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Global superlinear linearization schemes based on adaptive strategies for solving Richards’ equation

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 λ<inf>k</inf><sup>n</sup> via an error-correction approach to get fixed-point convergent sequences. We 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 Newton’s scheme (ECLN) to obtain global superlinear convergent schemes. 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. For the new schemes, we first apply the τ-adaptation to the classical methods (τ-Newton’s, τ-L-scheme, and τ-modified Picard). Next, we apply to the error-correction schemes (τ-AtS, τ-ALtS, τ-ALP, τ-ALN). Finally, we consider a combination of the L-scheme and the τ-adaptive Newton’s Method, mixing both methods (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 Newton’s scheme does not converge. Moreover, we include an example which uses the Gardner exponential nonlinearities, showing that L- and L2-schemes are as 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.