High-Order Entropy Stable Finite Difference Schemes for Nonlinear Conservation Laws: Finite Domains

Developing stable and robust high-order finite difference schemes requires mathematical formalism and appropriate methods of analysis. In this work, nonlinear entropy stability is used to derive provably stable high-order finite difference methods with formal boundary closures for conservation laws. Particular emphasis is placed on the entropy stability of the compressible Navier-Stokes equations. A newly derived entropy stable weighted essentially non-oscillatory finite difference method is used to simulate problems with shocks and a conservative, entropy stable, narrow-stencil finite difference approach is used to approximate viscous terms

Confronting the Alternate Level of Care (ALC) Crisis with a Multifaceted Policy Lens

Dual demands for increased provision of acute episodic care in hospital and chronic care in the community have contributed to an ALC crisis in Canadian hospitals, where large numbers of patients are boarded in acute-care beds rather than in environments more appropriate for their required level of care. Addressing this crisis will be one of the most profound challenges facing provincial health systems in Canada over the coming decades. This paper outlines the magnitude and complexity of confronting this growing crisis as well as defining a paradigm through which to explore and implement policy solutions along the entire continuum of challenges. ALC as an administrative designation aggregates diverse groups of patients covering a wide spectrum of demographic variables, medical diagnoses, social circumstances, discharge destinations and other characteristics, all of which can affect how and when ALC is coded. It is itself a significant challenge to collect consistent, accurate and adequately granular data to inform the design and implementation of policy reforms. With this in mind, a dominant association between advanced age and markedly higher ALC rates needs to be acknowledged and highlights that solutions to the ALC crisis will be significantly interwoven with addressing previously described challenges for the overall health system with an aging population. Clinically and operationally, ALC is a complex health-system issue that reflects and presents challenges from admission, throughout a patient’s hospital stay and after discharge. This paper outlines a holistic approach to categorizing policy interventions that address obstacles along this continuum, describing potential interventions in each phase. To achieve success, policy approaches must incorporate multi-faceted interventions into the overall context and systematize them to prevent, mitigate the burdens of, and improve the management of ALC

Developing a Family of Curves for the HEC-18 Scour Equation

Accurate pier scour predictions are essential to the safe and efficient design of bridge crossings. Current practice uses empirical formulas largely derived from laboratory experiments to predict local scour depth around single-bridge piers. The resulting formulas are hindered by insufficient consideration of scaling effects and hydrodynamic forces. When applied to full-scale designs, these formula deficiencies lead to excessive over prediction of scour depths and increased construction costs. In an effort to improve the predictive capabilities of the HEC-18 scour model, this work uses field-scale data and nonlinear regression to develop a family of equations optimized for various non-cohesive soil conditions. Improving the predictive capabilities of well-accepted equations saves scarce project dollars without sacrificing safety. To help improve acceptance of modified equations, this work strives to maintain the familiar form of the HEC- 18 equation. When compared to the HEC-18 local pier scour equation, this process reduced the mean square error of a validation data set while maintaining over prediction

Value-Based Mental Health Services for Youth and Families: The Role of Patient-Reported Outcome Measures in Youth Mental Health Services

In Alberta, the responsibility for youth mental-health is shared among three separate government ministries, compounding the challenge of determining the value of services delivered, especially from the youth’s own perspective. As a result, Alberta’s ability to measure service quality at the systems level is limited. Yet, given the short-term and long-term effects of poor mental health on youth, families, and society, there are clinical, moral, and economic imperatives for ensuring that all services provided are of the highest value possible.  Currently, Alberta is limited to estimating value mainly through quantitative measures focused on the cost of service delivery. However, value-based health-care services are measured as quality or outcomes for persons receiving health services in relation to the costs of delivering those services. One approach is to measure outcomes of youth receiving mental-health services from their own perspective to achieve value-based measurement of youth mental-health services.  Patient-reported outcome measures (PROMs) are questionnaires filled out by the persons receiving mental-health services themselves, and assess their self-reported health and well-being. PROMs have been shown to be important in evaluating the value of health-care services both at the individual and systems level.  At the individual level, PROMs allow patients and health-care providers to track progress over time. At the systems level, PROMs data can be compiled to evaluate trends between different sites or different health-care services or treatments over time, to help improve quality. Policy-makers can use these comparisons to help pinpoint which services offer the most value.  Given resource constraints, implementing PROMs province-wide in Alberta can improve the value of youth mental-health services at a time when they have become a matter of great urgency. Improving the quality and outcomes for youth and their families in the short term will deliver positive socioeconomic impacts in the future

Towards an Entropy Stable Spectral Element Framework for Computational Fluid Dynamics

Entropy stable (SS) discontinuous spectral collocation formulations of any order are developed for the compressible Navier-Stokes equations on hexahedral elements. Recent progress on two complementary efforts is presented. The first effort is a generalization of previous SS spectral collocation work to extend the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to tensor product Legendre-Gauss (LG) points. The LG and LGL point formulations are compared on a series of test problems. Although being more costly to implement, it is shown that the LG operators are significantly more accurate on comparable grids. Both the LGL and LG operators are of comparable efficiency and robustness, as is demonstrated using test problems for which conventional FEM techniques suffer instability. The second effort generalizes previous SS work to include the possibility of p-refinement at non-conforming interfaces. A generalization of existing entropy stability machinery is developed to accommodate the nuances of fully multi-dimensional summation-by-parts (SBP) operators. The entropy stability of the compressible Euler equations on non-conforming interfaces is demonstrated using the newly developed LG operators and multi-dimensional interface interpolation operators

Entropy Stable Staggered Grid Spectral Collocation for the Burgers' and Compressible Navier-Stokes Equations

Staggered grid, entropy stable discontinuous spectral collocation operators of any order are developed for Burgers' and the compressible Navier-Stokes equations on unstructured hexahedral elements. This generalization of previous entropy stable spectral collocation work [1, 2], extends the applicable set of points from tensor product, Legendre-Gauss-Lobatto (LGL) to a combination of tensor product Legendre-Gauss (LG) and LGL points. The new semi-discrete operators discretely conserve mass, momentum, energy and satisfy a mathematical entropy inequality for both Burgers' and the compressible Navier-Stokes equations in three spatial dimensions. They are valid for smooth as well as discontinuous flows. The staggered LG and conventional LGL point formulations are compared on several challenging test problems. The staggered LG operators are significantly more accurate, although more costly to implement. The LG and LGL operators exhibit similar robustness, as is demonstrated using test problems known to be problematic for operators that lack a nonlinearly stability proof for the compressible Navier-Stokes equations (e.g., discontinuous Galerkin, spectral difference, or flux reconstruction operators)

Boundary Closures for Fourth-order Energy Stable Weighted Essentially Non-Oscillatory Finite Difference Schemes

A general strategy exists for constructing Energy Stable Weighted Essentially Non Oscillatory (ESWENO) finite difference schemes up to eighth-order on periodic domains. These ESWENO schemes satisfy an energy norm stability proof for both continuous and discontinuous solutions of systems of linear hyperbolic equations. Herein, boundary closures are developed for the fourth-order ESWENO scheme that maintain wherever possible the WENO stencil biasing properties, while satisfying the summation-by-parts (SBP) operator convention, thereby ensuring stability in an L2 norm. Second-order, and third-order boundary closures are developed that achieve stability in diagonal and block norms, respectively. The global accuracy for the second-order closures is three, and for the third-order closures is four. A novel set of non-uniform flux interpolation points is necessary near the boundaries to simultaneously achieve 1) accuracy, 2) the SBP convention, and 3) WENO stencil biasing mechanics

High-Order Implicit-Explicit Multi-Block Time-stepping Method for Hyperbolic PDEs

This work seeks to explore and improve the current time-stepping schemes used in computational fluid dynamics (CFD) in order to reduce overall computational time. A high-order scheme has been developed using a combination of implicit and explicit (IMEX) time-stepping Runge-Kutta (RK) schemes which increases numerical stability with respect to the time step size, resulting in decreased computational time. The IMEX scheme alone does not yield the desired increase in numerical stability, but when used in conjunction with an overlapping partitioned (multi-block) domain significant increase in stability is observed. To show this, the Overlapping-Partition IMEX (OP IMEX) scheme is applied to both one-dimensional (1D) and two-dimensional (2D) problems, the nonlinear viscous Burger's equation and 2D advection equation, respectively. The method uses two different summation by parts (SBP) derivative approximations, second-order and fourth-order accurate. The Dirichlet boundary conditions are imposed using the Simultaneous Approximation Term (SAT) penalty method. The 6-stage additive Runge-Kutta IMEX time integration schemes are fourth-order accurate in time. An increase in numerical stability 65 times greater than the fully explicit scheme is demonstrated to be achievable with the OP IMEX method applied to 1D Burger's equation. Results from the 2D, purely convective, advection equation show stability increases on the order of 10 times the explicit scheme using the OP IMEX method. Also, the domain partitioning method in this work shows potential for breaking the computational domain into manageable sizes such that implicit solutions for full three-dimensional CFD simulations can be computed using direct solving methods rather than the standard iterative methods currently used

Entropy Stable Spectral Collocation Schemes for the Navier-Stokes Equations: Discontinuous Interfaces

Nonlinear entropy stability and a summation-by-parts framework are used to derive provably stable, polynomial-based spectral collocation methods of arbitrary order. The new methods are closely related to discontinuous Galerkin spectral collocation methods commonly known as DGFEM, but exhibit a more general entropy stability property. Although the new schemes are applicable to a broad class of linear and nonlinear conservation laws, emphasis herein is placed on the entropy stability of the compressible Navier-Stokes equations

Discretely Conservative Finite-Difference Formulations for Nonlinear Conservation Laws in Split Form: Theory and Boundary Conditions

Simulations of nonlinear conservation laws that admit discontinuous solutions are typically restricted to discretizations of equations that are explicitly written in divergence form. This restriction is, however, unnecessary. Herein, linear combinations of divergence and product rule forms that have been discretized using diagonal-norm skew-symmetric summation-by-parts (SBP) operators, are shown to satisfy the sufficient conditions of the Lax-Wendroff theorem and thus are appropriate for simulations of discontinuous physical phenomena. Furthermore, special treatments are not required at the points that are near physical boundaries (i.e., discrete conservation is achieved throughout the entire computational domain, including the boundaries). Examples are presented of a fourth-order, SBP finite-difference operator with second-order boundary closures. Sixth- and eighth-order constructions are derived, and included in E. Narrow-stencil difference operators for linear viscous terms are also derived; these guarantee the conservative form of the combined operator