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Numerical methods in modern computing architectures for multidomain problems

Vilanakis Nikolaos

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Summary

Multidomain problems arise in many scientific fields and require advanced mathematical models and computational methods to interpret physical phenomena. In geophysics, a fundamental problem is the exploration of the subsurface using electromagnetic fields. Electromagnetic sounding methods aim to analyze the subsurface and identify geologic formations of high interest. The field process involves emitting a primary electromagnetic field at a fixed frequency into the environment using an appropriate transmitter-receiver device. This field interacts with subsurface materials of varying physical properties, such as specific electrical conductivity, and records the response of the secondary field. Field measurements and analysis of the secondary field contribute to iteratively adjusting and improving a (initially hypothetical) earth model in another crucial phase of the process, solving the forward problem. The mathematical approach to the forward problem is based on Maxwell’s equations, which describe the relationship between electric and magnetic fields. Their combination, aimed at decoupling the fields, leads to a Helmholtz-type differential equation for the intensity of the electric field. Solving this equation to estimate the intensity of the electric field in three-dimensional models with inhomogeneous materials is computationally demanding due to the complexity of the problem and must also be precise. This study presents a new solver for this equation, employing a compact fourth-order accuracy finite-difference scheme on a staggered grid to approximate the components of the electric field intensity in the frequency domain for a computational model of a three-dimensional half-space. For the solution of the resulting from the discretization linear system, methods such as the BiCGSTAB iterative method and Cyclic Reduction are being used. The solver is designed to be implemented on multicore computational systems and parallel architectures using the OpenMP standard, with low memory requirements, and is extensible to grid-based architectures to generate approximations for multiple transmitter positions. The results include extensive tests using varying transmitter heights and different subsurface electrical conductivity values in the half-space model.

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