A method for modeling non-Newtonian fluids (dilatants and pseudoplastics) by a power law under the Godunov-Peshkov-Romenski model is presented, along with a new numerical scheme for solving this system. The scheme is also modified to solve the corresponding system for power-law elastoplastic solids.The scheme is based on a temporal operator splitting, with the homogeneous system solved using a finite volume method based on a WENO reconstruction, and the temporal ODEs solved using an analytical approximate solution. The method is found to perform favorably against problems with known exact solutions, and numerical solutions published in the open literature. It is simple to implement, and to the best of the authors' knowledge it is currently the only method for solving this modified version of the GPR model.

No abstract

A new second-order numerical scheme based on an operator splitting is proposed for the Godunov-Peshkov-Romenski model of continuum mechanics. The homogeneous part of the system is solved with a finite volume method based on a WENO reconstruction, and the temporal ODEs are solved using some analytic results presented here. Whilst it is not possible to attain arbitrary-order accuracy with this scheme (as with ADER-WENO schemes used previously), the attainable order of accuracy is often sufficient, and solutions are computationally cheap when compared with other available schemes. The new scheme is compared with an ADER-WENO scheme for various test cases, and a convergence study is undertaken to demonstrate its order of accuracy.

A framework for simulating the interactions between multiple different continua is presented. Each constituent material is governed by the same set of equations, differing only in terms of their equations of state and strain dissipation functions. The interfaces between any combination of fluids, solids, and vacuum are handled by a new Riemann Ghost Fluid Method, which is agnostic to the type of material on either side (depending only on the desired boundary conditions).The Godunov-Peshkov-Romenski (GPR) model is used for modeling the continua (having recently been used to solve a range of problems involving Newtonian and non-Newtonian fluids, and elastic and elastoplastic solids), and this study represents a novel approach for handling multimaterial problems under this model.The resulting framework is simple, yet capable of accurately reproducing a wide range of different physical scenarios. It is demonstrated here to accurately reproduce analytical results for known Riemann problems, and to produce expected results in other cases, including some featuring heat conduction across interfaces, and impact-induced deformation and detonation of combustible materials. The framework thus has the potential to streamline development of simulation software for scenarios involving multiple materials and phases of matter, by reducing the number of different systems of equations that require solvers, and cutting down on the amount of theoretical work required to deal with the interfaces between materials.

Hyperbolic systems of PDEs can be solved to arbitrary orders of accuracy by using the ADER Finite Volume method. These PDE systems may be non-conservative and non-homogeneous, and contain stiff source terms. ADER-FV requires a spatio-temporal polynomial reconstruction of the data in each spacetime cell, at each time step. This reconstruction is obtained as the root of a nonlinear system, resulting from the use of a Galerkin method. It was proved in Jackson [7] that for traditional choices of basis polynomials, the eigenvalues of certain matrices appearing in these nonlinear systems are always 0, regardless of the number of spatial dimensions of the PDEs or the chosen order of accuracy of the ADER-FV method. This guarantees fast convergence to the Galerkin root for certain classes of PDEs.In Montecinos and Balsara [9] a new, more efficient class of basis polynomials for the one-dimensional ADER-FV method was presented. This new class of basis polynomials, originally presented for conservative systems, is extended to multidimensional, non-conservative systems here, and the corresponding property regarding the eigenvalues of the Galerkin matrices is proved.