Arbitrary order dissipative and conservative Hermite methods for the scalar wave equation are presented. Both methods use (m + 1) d degrees of freedom per node for the displacement in d-dimensions; the dissipative and conservative methods achieve orders of accuracy (2m − 1) and 2m, respectively. Stability and error analyses as well as implementation strategies for accelerators are also given.1. Introduction. We construct, analyze, and test arbitrary order dissipative and conservative Hermite methods for the scalar wave equation in a medium with constant speed of sound c. The degrees-of-freedom for Hermite methods are tensor-product Taylor polynomials of degree m in each coordinate centered at the nodes of Cartesian grids, staggered in time. The dissipative method achieves space-time accuracy of order 2m − 1, while the conservative method has space-time order 2m. Besides their high order of accuracy in both space and time combined, they have the special feature that they are stable for c∆t ≤ h, for all orders of accuracy. This is significantly better than standard high-order element methods. Moreover, the large time steps are purely local to each cell, minimizing communication and storage requirements.Our primary interest in these schemes are as highly efficient building blocks in hybrid methods where most of the mesh can be taken to be rectilinear and where geometry is handled by more flexible (but less efficient) methods close to physical boundaries. In this work we restrict our consideration to square geometries with boundary conditions of Dirichlet, Neumann or periodic type, where boundary conditions are simple to apply. In previous work [7] we considered this type of hybridization of the Hermite methods for first order system proposed in [11] with nodal discontinuous Galerkin (dG) methods. For wave equations in second order form we envision a similar hybridization where the geometry is handled by, for example, our recently developed discontinuous Galerkin methods for wave equations in second order form [1]. Our dG method has the property that, based on the choice of numerical flux, it is either dissipative or conservative.We provide optimal stability and convergence results for both the conservative and dissipative method for one dimensional periodic domains. The analysis for the dissipative method follows the analysis for first order systems [11] but here it is based on the energy of the wave equation v 2 + |∇u| 2 dx. A difference compared to [11] is that we require that the (polynomial) approximation spaces of the velocity, v, and displacement, u to differ by one degree. Its extension to higher space dimensions, however, does not follow in a straightforward way from the Hermite method for first-order systems, requiring a specialized interpolation scheme to achieve order-independent stability at CFL one.The analysis of the conservative method is new and quite different from that of [11]. Additionally, the analysis is done by introducing what we denote conserved variables, this simplifies the analysis conside...
We explore a framework to model the dose response of allosteric multisite phosphorylation proteins using a single auxiliary variable. This reduction can closely replicate the steady state behavior of detailed multisite systems such as the Monod-Wyman-Changeux allosteric model or rule-based models. Optimal ultrasensitivity is obtained when the activation of an allosteric protein by its individual sites is concerted and redundant. The reduction makes this framework useful for modeling and analyzing biochemical systems in practical applications, where several multisite proteins may interact simultaneously. As an application we analyze a newly discovered checkpoint signaling pathway in budding yeast, which has been proposed to measure cell growth by monitoring signals generated at sites of plasma membrane growth. We show that the known components of this pathway can form a robust hysteretic switch. In particular, this system incorporates a signal proportional to bud growth or size, a mechanism to read the signal, and an all-or-none response triggered only when the signal reaches a threshold indicating that sufficient growth has occurred.
The Hermite methods of Goodrich, Hagstrom, and Lorenz (2006) use Hermite interpolation to construct high order numerical methods for hyperbolic initial value problems. The structure of the method has several favorable features for parallel computing. In this work, we propose algorithms that take advantage of the many-core architecture of Graphics Processing Units. The algorithm exploits the compact stencil of Hermite methods and uses data structures that allow for efficient data load and stores. Additionally the highly localized evolution operator of Hermite methods allows us to combine multi-stage time-stepping methods within the new algorithms incurring minimal accesses of global memory. Using a scalar linear wave equation, we study the algorithm by considering Hermite interpolation and evolution as individual kernels and alternatively combined them into a monolithic kernel. For both approaches we demonstrate strategies to increase performance. Our numerical experiments show that although a two kernel approach allows for better performance on the hardware, a monolithic kernel can offer a comparable time to solution with less global memory usage.
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