Novel numerical methods for time-space fractional reaction diffusion equations in two dimensions

We consider time-space fractional reaction diffusion equations in two dimensions. This equation is obtained from the standard reaction diffusion equation by replacing the first order time derivative with the Caputo fractional derivative, and the second order space derivatives with the fractional Laplacian. Using the matrix transfer technique proposed by Ilic, Liu, Turner and Anh [Fract. Calc. Appl. Anal., 9:333--349, 2006] and the numerical solution strategy used by Yang, Turner, Liu, and Ilic [SIAM J. Scientific Computing, 33:1159--1180, 2011], the solution of the time-space fractional reaction diffusion equations in two dimensions can be written in terms of a matrix function vector product $f(A)b$ at each time step, where $A$ is an approximate matrix representation of the standard Laplacian. We use the finite volume method over unstructured triangular meshes to generate the matrix $A$, which is therefore non-symmetric. However, the standard Lanczos method for approximating $f(A)b$ requires that $A$ is symmetric. We propose a simple and novel transformation in which the standard Lanczos method is still applicable to find $f(A)b$, despite the loss of symmetry. Numerical results are presented to verify the accuracy and efficiency of our newly proposed numerical solution strategy.

Pulling apart the mid to late Cenozoic magmatic record of the Gulf of California : is there a Comondú arc?

The composition of the lithosphere can be fundamentally altered by long-lived subduction processes such that subduction-modified lithosphere can survive for 100's Myrs. Incorrect petrotectonic interpretations result when spatial-temporal-compositional trends of, and source contributions to, magmatism are not properly considered. Western Mexico has had protracted Cenozoic magmatism developed mostly in-board of active oceanic plate subduction beneath western North America. A broad range of igneous compositions from basalt to high-silica rhyolite were erupted with intermediate to silicic compositions in particular, showing calc-alkaline and other typical subduction-related geochemical signatures. A major Oligocene rhyolitic ignimbrite “flare-up” (>300,000 km3) switched to a bimodal volcanic phase in the Early Miocene (~100,000 km3), associated with distributed extension and opening of numerous grabens. Extension became more focussed ~18 Ma resulting in localised volcanic activity along the future site of the Gulf of California. This localised volcanism (known as the Comondú “arc”) was dominantly effusive and andesite-dacite in composition. Past tectonic interpretations of Comondú-age volcanism may have been incorrect as these regional temporal-compositional changes are alternatively interpreted as a result of increased mixing of mantle-derived basaltic and crust-derived rhyolitic magmas in an active rift environment rather than fluid flux melting of the mantle wedge above the subducting Guadalupe Plate.