Tempus is a software framework for the time integration of ordinary differential equations (ODEs)... more Tempus is a software framework for the time integration of ordinary differential equations (ODEs), partial differential equations (PDEs), and differential-algebraic equations (DAEs). Examples of time-integration methods available are Forward Euler, Backward Euler, Explicit Runge-Kutta and Implicit Runge-Kutta methods. Tempus can be used from small systems of equations (e.g., single ODEs for the time evolution of plasticity models) to large-scale transient simulations requiring exascale computing (e.g., flow fields around reentry vehicles and magneto-hydrodynamics). Tempus will be released as part of Trilinos under a BSD open source license.
Version supporting the draft of "Embedded error estimation and adaptive step-size control fo... more Version supporting the draft of "Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods"
Diabetes is a collection of diseases marked by high levels of glucose in the blood. The condition... more Diabetes is a collection of diseases marked by high levels of glucose in the blood. The condition results from defects in insulin production or function, which are activities performed by the pancreas. Within the endocrine system of the pancreas lie clusters of cells called islets. Each islet is composed of four different cells, the most prevalent of which being the beta cell. The main function of beta cells is to secrete insulin in response to blood glucose levels. As a result, the behavior of these cells is an issue of ongoing interest in diabetes research. Our research aims to take the next step in implementing the mathematical model governing beta cells by continuing the development of a computational islet. The mechanisms of insulin secretion within beta cells can be modeled with a set of deterministic ordinary differential equations. Considering cell dynamics of a cube of individual heterogeneous cells, the key parameters influencing the time evolution include ionic fluxes, ca...
Proceedings of the Practice and Experience in Advanced Research Computing 2017 on Sustainability, Success and Impact, 2017
High order strong stability preserving (SSP) time discretizations have been extensively used with... more High order strong stability preserving (SSP) time discretizations have been extensively used with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. Explicit SSP Runge--Kutta methods exist only up to fourth order, and implicit SSP Runge--Kutta methods exist only up to sixth order. When solving linear autonomous problems, the order conditions simplify and this order barrier is lifted: SSP Runge--Kutta methods of any linear order exist. In this work, we extend the concept of varying orders of accuracy for linear and non linear components to the class of implicit-explicit (IMEX) Runge--Kutta methods methods. We formulate an optimization problem for implicit-explicit (IMEX) SSP Runge--Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge--Kutta methods of orders plin = 3, 4, 6 as well as optimized IMEX SSP Runge--Kutta pairs that have high linear order and nonlinear orders p = 2, ...
We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Ku... more We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.
Tempus is a software framework for the time integration of ordinary differential equations (ODEs)... more Tempus is a software framework for the time integration of ordinary differential equations (ODEs), partial differential equations (PDEs), and differential-algebraic equations (DAEs). Examples of time-integration methods available are Forward Euler, Backward Euler, Explicit Runge-Kutta and Implicit Runge-Kutta methods. Tempus can be used from small systems of equations (e.g., single ODEs for the time evolution of plasticity models) to large-scale transient simulations requiring exascale computing (e.g., flow fields around reentry vehicles and magneto-hydrodynamics). Tempus will be released as part of Trilinos under a BSD open source license.
Version supporting the draft of "Embedded error estimation and adaptive step-size control fo... more Version supporting the draft of "Embedded error estimation and adaptive step-size control for optimal explicit strong stability preserving Runge–Kutta methods"
Diabetes is a collection of diseases marked by high levels of glucose in the blood. The condition... more Diabetes is a collection of diseases marked by high levels of glucose in the blood. The condition results from defects in insulin production or function, which are activities performed by the pancreas. Within the endocrine system of the pancreas lie clusters of cells called islets. Each islet is composed of four different cells, the most prevalent of which being the beta cell. The main function of beta cells is to secrete insulin in response to blood glucose levels. As a result, the behavior of these cells is an issue of ongoing interest in diabetes research. Our research aims to take the next step in implementing the mathematical model governing beta cells by continuing the development of a computational islet. The mechanisms of insulin secretion within beta cells can be modeled with a set of deterministic ordinary differential equations. Considering cell dynamics of a cube of individual heterogeneous cells, the key parameters influencing the time evolution include ionic fluxes, ca...
Proceedings of the Practice and Experience in Advanced Research Computing 2017 on Sustainability, Success and Impact, 2017
High order strong stability preserving (SSP) time discretizations have been extensively used with... more High order strong stability preserving (SSP) time discretizations have been extensively used with spatial discretizations with nonlinear stability properties for the solution of hyperbolic PDEs. Explicit SSP Runge--Kutta methods exist only up to fourth order, and implicit SSP Runge--Kutta methods exist only up to sixth order. When solving linear autonomous problems, the order conditions simplify and this order barrier is lifted: SSP Runge--Kutta methods of any linear order exist. In this work, we extend the concept of varying orders of accuracy for linear and non linear components to the class of implicit-explicit (IMEX) Runge--Kutta methods methods. We formulate an optimization problem for implicit-explicit (IMEX) SSP Runge--Kutta methods and find implicit methods with large linear stability regions that pair with known explicit SSP Runge--Kutta methods of orders plin = 3, 4, 6 as well as optimized IMEX SSP Runge--Kutta pairs that have high linear order and nonlinear orders p = 2, ...
We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Ku... more We construct a family of embedded pairs for optimal strong stability preserving explicit Runge-Kutta methods of order $2 \leq p \leq 4$ to be used to obtain numerical solution of spatially discretized hyperbolic PDEs. In this construction, the goals include non-defective methods, large region of absolute stability, and optimal error measurement as defined in [5,19]. The new family of embedded pairs offer the ability for strong stability preserving (SSP) methods to adapt by varying the step-size based on the local error estimation while maintaining their inherent nonlinear stability properties. Through several numerical experiments, we assess the overall effectiveness in terms of precision versus work while also taking into consideration accuracy and stability.
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Papers by Sidafa Conde