Mathematical methods for geophysical problems
Numerical modeling of geodynamic processes is recognized as a challenging computational problem which requires use of advanced computational techniques and development of powerful numerical tools. One of the major challenges concerns solving of the inertia-free Stokes equation coupled to the incompressible continuity equation in a combination with strong viscosity variations in the computational domain. Consequently, benchmarking of numerical codes against analytical and numerical solutions constrained for various mechanical and thermomechanical Stokes flow problems is a common practice in computational geodynamics.
Available analytical and numerical solutions are mostly two-dimensional. These solutions are constrained for a number of well defined model setups, which are of potential significance for various situations which numerical codes may face during real geodynamic simulations. Availability and broad range of 2D and 3D benchmark solutions are, therefore, critical for the development and testing of the next generation of numerical geodynamic modeling software which aims to combine rheological complexity of constitutive laws with adaptive grid resolution to on both global and regional scales.
The main goal of the project is to significantly expand availability of benchmark solutions for both 2D and 3D variable viscosity Stokes flows. In contrast to previous studies, we prefer not to start from any prescribed model setups but rather derive general analytical solutions, which are potentially suitable for generating a broad range of test problems. We derive generalized solutions for incompressible Stokes problems with (a) linearly and (b) exponentially variable viscosity. In the following we demonstrate how these generalized solutions can be converted into 2D and 3D test problems suitable for benchmarking numerical codes. Finally, based on the obtained benchmark problems , we show examples of numerical convergence tests for staggered-grid discretizations schemes.
Figure 1. Distribution of vx, vy and P; 2D case, linearly varying viscosity, low viscosity contrast.
Figure 2. Logarithm of the relative error via logarithm of the grid step;