Von Neumann Stability Analysis of DG-Like and PNPM-Like Schemes for PDEs with Globally Curl-Preserving Evolution of Vector Fields

This paper examines a class of involution-constrained PDEs where some part of the PDE system evolves a vector field whose curl remains zero or grows in proportion to specified source terms. Such PDEs are referred to as curl-free or curl-preserving, respectively. They arise very frequently in equations for hyperelasticity and compressible multiphase flow, in certain formulations of general relativity and in the numerical solution of Schrödinger's equation. Experience has shown that if nothing special is done to account for the curl-preserving vector field, it can blow up in a finite amount of simulation time. In this paper, we catalogue a class of DG-like schemes for such PDEs. To retain the globally curl-free or curl-preserving constraints, the components of the vector field, as well as their higher moments, must be collocated at the edges of the mesh. They are updated using potentials collocated at the vertices of the mesh. The resulting schemes: (i) do not blow up even after very long integration times, (ii) do not need any special cleaning treatment, (iii) can operate with large explicit timesteps, (iv) do not require the solution of an elliptic system and (v) can be extended to higher orders using DG-like methods. The methods rely on a special curl-preserving reconstruction and they also rely on multidimensional upwinding. The Galerkin projection, highly crucial to the design of a DG method, is now conducted at the edges of the mesh and yields a weak form update that uses potentials obtained at the vertices of the mesh with the help of a multidimensional Riemann solver. A von Neumann stability analysis of the curl-preserving methods is conducted and the limiting CFL numbers of this entire family of methods are catalogued in this work. The stability analysis confirms that with the increasing order of accuracy, our novel curl-free methods have superlative phase accuracy while substantially reducing dissipation. We also show that PNPM-like methods, which only evolve the lower moments while reconstructing the higher moments, retain much of the excellent wave propagation characteristics of the DG-like methods while offering a much larger CFL number and lower computational complexity. The quadratic energy preservation of these methods is also shown to be excellent, especially at higher orders. The methods are also shown to be curl-preserving over long integration times.

A discontinuous Galerkin model for fluorescence loss in photobleaching of intracellular polyglutamine protein aggregates

Background

Intracellular phase separation and aggregation of proteins with extended poly-glutamine (polyQ) stretches are hallmarks of various age-associated neurodegenerative diseases. Progress in our understanding of such processes heavily relies on quantitative fluorescence imaging of suitably tagged proteins. Fluorescence loss in photobleaching (FLIP) is particularly well-suited to study the dynamics of protein aggregation in cellular models of Chorea Huntington and other polyQ diseases, as FLIP gives access to the full spatio-temporal profile of intensity changes in the cell geometry. In contrast to other methods, also dim aggregates become visible during time evolution of fluorescence loss in cellular compartments. However, methods for computational analysis of FLIP data are sparse, and transport models for estimation of transport and diffusion parameters from experimental FLIP sequences are missing.

Results

In this paper, we present a computational method for analysis of FLIP imaging experiments of intracellular polyglutamine protein aggregates also called inclusion bodies (IBs). By this method, we can determine the diffusion constant and nuclear membrane transport coefficients of polyQ proteins as well as the exchange rates between aggregates and the cytoplasm. Our method is based on a reaction-diffusion multi-compartment model defined on a mesh obtained by segmentation of the cell images from the FLIP sequence. The discontinuous Galerkin (DG) method is used for numerical implementation of our model in FEniCS, which greatly reduces the computing time. The method is applied to representative experimental FLIP sequences, and consistent estimates of all transport parameters are obtained.

Conclusions

By directly estimating the transport parameters from live-cell image sequences using our new computational FLIP approach surprisingly fast exchange dynamics of mutant Huntingtin between cytoplasm and dim IBs could be revealed. This is likely relevant also for other polyQ diseases. Thus, our method allows for quantifying protein dynamics at different stages of the protein aggregation process in cellular models of neurodegeneration.

Electronic supplementary material

The online version of this article (10.1186/s13628-018-0046-0) contains supplementary material, which is available to authorized users.

EXPLICIT LEAST-DEGREE BOUNDARY FILTERS FOR DISCONTINUOUS GALERKIN

Convolving the output of Discontinuous Galerkin (DG) computations using spline filters can improve both smoothness and accuracy of the output. At domain boundaries, these filters have to be one-sided for non-periodic boundary conditions. Recently, position-dependent smoothness-increasing accuracy-preserving (PSIAC) filters were shown to be a superset of the well-known one-sided RLKV and SRV filters. Since PSIAC filters can be formulated symbolically, PSIAC filtering amounts to forming linear products with local DG output and so offers a more stable and efficient implementation. The paper introduces a new class of PSIAC filters NP₀ that have small support and are piecewise constant. Extensive numerical experiments for the canonical hyperbolic test equation show NP₀ filters outperform the more complex known boundary filters. NP₀ filters typically reduce the L^∞ error in the boundary region below that of the interior where optimally superconvergent symmetric filters of the same support are applied. NP₀ filtering can be implemented as forming linear combinations of the data with short rational weights. Exact derivatives of the convolved output are easy to compute.

Higher-order accurate space-time schemes for computational astrophysics-Part I: finite volume methods

As computational astrophysics comes under pressure to become a precision science, there is an increasing need to move to high accuracy schemes for computational astrophysics. The algorithmic needs of computational astrophysics are indeed very special. The methods need to be robust and preserve the positivity of density and pressure. Relativistic flows should remain sub-luminal. These requirements place additional pressures on a computational astrophysics code, which are usually not felt by a traditional fluid dynamics code. Hence the need for a specialized review. The focus here is on weighted essentially non-oscillatory (WENO) schemes, discontinuous Galerkin (DG) schemes and PNPM schemes. WENO schemes are higher order extensions of traditional second order finite volume schemes. At third order, they are most similar to piecewise parabolic method schemes, which are also included. DG schemes evolve all the moments of the solution, with the result that they are more accurate than WENO schemes. PNPM schemes occupy a compromise position between WENO and DG schemes. They evolve an Nth order spatial polynomial, while reconstructing higher order terms up to Mth order. As a result, the timestep can be larger. Time-dependent astrophysical codes need to be accurate in space and time with the result that the spatial and temporal accuracies must be matched. This is realized with the help of strong stability preserving Runge-Kutta schemes and ADER (Arbitrary DERivative in space and time) schemes, both of which are also described. The emphasis of this review is on computer-implementable ideas, not necessarily on the underlying theory.