A New Family of High Order Unstructured MOOD and ADER Finite Volume Schemes for Multidimensional Systems of Hyperbolic Conservation Laws

In this paper,we investigate the coupling of the Multi-dimensional Optimal Order Detection(MOOD)method and the Arbitrary high order DERivatives(ADER)approach in order to design a new high order accurate,robust and computationally efficient Finite Volume(FV)scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions,respectively.The Multi-dimensional Optimal Order Detection(MOOD)method for 2D and 3D geometries has been introduced in a recent series of papers for mixed unstructured meshes.It is an arbitrary high-order accurate Finite Volume scheme in space,using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.In the following work,the time discretization is performed with an elegant and efficient one-step ADER procedure.Doing so,we retain the good properties of the MOOD scheme,that is to say the optimal high-order of accuracy is reached on smooth solutions,while spurious oscillations near singularities are prevented.The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D,but it also increases the stability of the overall scheme.A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost,robustness,accuracy and efficiency.The main finding of this paper is that the combination of ADER with MOOD generally outperforms the one of ADER and WENO either because at given accuracy MOOD is less expensive(memory and/or CPU time),or because it is more accurate for a given grid resolution.A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws:the Euler equations of compressible gas dynamics,the classical equations of ideal magneto-Hydrodynamics(MHD)and finally the relativistic MHD equations(RMHD),which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.All tests are run on genuinely unstructured grids composed of simplex elements.

Parallel Algorithms and Software for Nuclear,Energy,and Environmental Applications.Part Ⅱ:Multiphysics Software

This paper is the second part of a two part sequence on multiphysics algorithms and software.The first[1]focused on the algorithms;this part treats the multiphysics software framework and applications based on it.Tight coupling is typically designed into the analysis application at inception,as such an application is strongly tied to a composite nonlinear solver that arrives at the final solution by treating all equations simultaneously.The applicationmust also take care tominimize both time and space error between the physics,particularly if more than one mesh representation is needed in the solution process.This paper presents an application framework that was specifically designed to support tightly coupled multiphysics analysis.The Multiphysics Object Oriented Simulation Environment(MOOSE)is based on the Jacobian-freeNewton-Krylov(JFNK)method combined with physics-based preconditioning to provide the underlying mathematical structure for applications.The report concludes with the presentation of a host of nuclear,energy,and environmental applications that demonstrate the efficacy of the approach and the utility of a well-designed multiphysics framework.

Parallel Algorithms and Software for Nuclear,Energy,and Environmental Applications.Part Ⅰ:Multiphysics Algorithms

There is a growing trend within energy and environmental simulation to consider tightly coupled solutions to multiphysics problems.This can be seen in nuclear reactor analysis where analysts are interested in coupled flow,heat transfer and neutronics,and in nuclear fuel performance simulation where analysts are interested in thermomechanics with contact coupled to species transport and chemistry.In energy and environmental applications,energy extraction involves geomechanics,flow through porous media and fractured formations,adding heat transport for enhanced oil recovery and geothermal applications,and adding reactive transport in the case of applications modeling the underground flow of contaminants.These more ambitious simulations usually motivate some level of parallel computing.Many of the physics coupling efforts to date utilize simple code coupling or first-order operator splitting,often referred to as loose coupling.While these approaches can produce answers,they usually leave questions of accuracy and stability unanswered.Additionally,the different physics often reside on distinct meshes and data are coupled via simple interpolation,again leaving open questions of stability and accuracy.∗Corresponding author.Email addresses:Derek.Gaston@inl.gov(D.Gaston),This paper is the first part of a two part sequence on multiphysics algorithms and software.Part I examines the importance of accurate time and space integration and that the degree of coupling used for the solution should match the requirements of the simulation.It then discusses the preconditioned Jacobian-free Newton Krylov solution algorithm that is used for both multiphysics and multiscale solutions.Part II[1]presents the software framework;the Multiphysics Object Oriented Simulation Environment(MOOSE)and discusses applications based on it.