Performance of explicit and IMEX MRI multirate methods on complex reactive flow problems within modern parallel adaptive structured grid frameworks

Large-scale multiphysics simulations are computationally challenging due to the coupling of multiple processes with widely disparate time scales. The advent of exascale computing systems exacerbates these challenges, since these enable ever increasing size and complexity. Recently, there has been renewed interest in developing multirate methods as a means to handle the large range of time scales, as these methods may afford greater accuracy and efficiency than more traditional approaches of using IMEX and low-order operator splitting schemes. However, there have been few performance studies that compare different classes of multirate integrators on complex application problems. We study the performance of several newly developed multirate infinitesimal (MRI) methods, implemented in the SUNDIALS solver package, on two reacting flow model problems built on structured mesh frameworks. The first model revisits the work of Emmet et al. (2014) on a compressible reacting flow problem with complex chemistry that is implemented using BoxLib but where we now include comparisons between a new explicit MRI scheme with the multirate spectral deferred correction (SDC) methods in the original paper. The second problem uses the same complex chemistry as the first problem, combined with a simplified flow model, but run at a large spatial scale where explicit methods become infeasible due to stability constraints. Two recently developed implicit-explicit MRI multirate methods are tested. These methods rely on advanced features of the AMReX framework on which the model is built, such as multilevel grids and multilevel preconditioners. The results from these two problems show that MRI multirate methods can offer significant performance benefits on complex multiphysics application problems and that these methods may be combined with advanced spatial discretization to compound the advantages of both

Implicit-explicit multirate infinitesimal GARK methods

This work focuses on the development of a new class of high-order accurate methods for multirate time integration of systems of ordinary differential equations. Unlike other recent work in this area, the proposed methods support mixed implicit-explicit (IMEX) treatment of the slow time scale. In addition to allowing this slow time scale flexibility, the proposed methods utilize a so-called `infinitesimal' formulation for the fast time scale through definition of a sequence of modified `fast' initial-value problems, that may be solved using any viable algorithm. We name the proposed class as implicit-explicit multirate infinitesimal generalized-structure additive Runge--Kutta (IMEX-MRI-GARK) methods. In addition to defining these methods, we prove that they may be viewed as specific instances of generalized-structure additive Runge--Kutta (GARK) methods, and derive a set of order conditions on the IMEX-MRI-GARK coefficients to guarantee both third and fourth order accuracy for the overall multirate method. Additionally, we provide three specific IMEX-MRI-GARK methods, two of order three and one of order four. We conclude with numerical simulations on two multirate test problems, demonstrating the methods' predicted convergence rates and comparing their efficiency against both legacy IMEX multirate schemes and recent third and fourth order implicit MRI-GARK methods

Modelação e Simulação Eletrotérmica de Circuitos e Sistemas Eletrónicos

Esta dissertação insere-se na área da modelação e simulação eletrotérmica de circuitos eletrónicos que contêm MOSFETs de potência. Visa essencialmente o estudo da aplicabilidade de ferramentas computacionais inovadoras que consigam simular, de forma eficiente, circuitos que operem em múltiplas escalas temporais, como é o caso da simulação elétrica e térmica conjunta. A simulação eletrotérmica de um componente eletrónico cujo funcionamento depende fortemente da temperatura necessita que, durante o seu período de operação, se conheça com rigor o valor da temperatura em vários pontos do seu interior, de modo a se poder estimar o seu comportamento. O modelo do MOSFET utilizado, baseado em modelos SPICE, é um modelo eletrotérmico contínuo, que permite que a temperatura de funcionamento seja atualizada dinamicamente durante o processo de simulação. Em conjunto com o modelo do MOSFET são também adotados nesta dissertação modelos de propagação térmica baseados em linhas de transmissão de calor. Para se poder tirar o proveito dos diferentes ritmos de evolução temporal existentes entre as variáveis de estado elétricas e térmicas, são utilizadas algumas técnicas numéricas avançadas baseadas em esquemas Runge-Kutta multi-ritmo. Nesta dissertação é dada especial atenção ao método de ordem 2(3). O desempenho deste método numérico é avaliado em dois exemplos de aplicação ilustrativos, com resultados bastante interessantes. Através da análise comparativa entre os resultados obtidos com os métodos numéricos convencionais presentes nos simuladores SPICE e o método proposto, é possível constatar ganhos significativos em termos de poupança de esforço computacional