An Unconditional Positivity-Preserving Difference Scheme for Models of Cancer Migration and Invasion

In this paper, we consider models of cancer migration and invasion, which consist of two nonlinear parabolic equations (one of the convection–diffusion reaction type and the other of the diffusion–reaction type) and an additional nonlinear ordinary differential equation. The unknowns represent concentrations or densities that cannot be negative. Widely used approximations, such as difference schemes, can produce negative solutions because of truncation errors and can become unstable. We propose a new difference scheme that guarantees the positivity of the numerical solution for arbitrary mesh step sizes. It has explicit and fast performance even for nonlinear reaction terms that consist of sums of positive and negative functions. The numerical examples illustrate the simplicity and efficiency of the method. A numerical simulation of a model of cancer migration is also discussed.

Variable Step Size Adams Methods for BSDEs

For backward stochastic differential equations (BSDEs), we construct variable step size Adams methods by means of Itô–Taylor expansion, and these schemes are nonlinear multistep schemes. It is deduced that the conditions of local truncation errors with respect to Y and Z reach high order. The coefficients in the numerical methods are inferred and bounded under appropriate conditions. A necessary and sufficient condition is given to judge the stability of our numerical schemes. Moreover, the high-order convergence of the schemes is rigorously proved. The numerical illustrations are provided.

Binaural Reproduction Based on Bilateral Ambisonics

Binaural reproduction of high-quality spatial sound has gained considerable interest with the recent technology developments in virtual and augmented reality. The reproduction of binaural signals in the Spherical-Harmonics (SH) domain using Ambisonics is now a well-established methodology, with flexible binaural processing realized using SH representations of the sound-field and the Head-Related Transfer Function (HRTF). However, in most practical cases, the binaural reproduction is order-limited, which introduces truncation errors that have a detrimental effect on the perception of the reproduced signals, mainly due to the truncation of the HRTF. Recently, it has been shown that manipulating the HRTF phase component, by ear-alignment, significantly reduces its effective SH order while preserving its phase information, which may be beneficial for alleviating the above detrimental effect. Incorporating the ear-aligned HRTF into the binaural reproduction process has been suggested by using Bilateral Ambisonics, which is an Ambisonics representation of the sound-field formulated at the two ears. While this method imposes challenges on acquiring the sound-field, and specifically, on applying head-rotations, it leads to a significant reduction in errors caused by the limited-order reproduction, which yields a substantial improvement in the perceived binaural reproduction quality even with first order SH.

Uncertainty propagation and truncation errors in LPT kinematics

Lagrangian Particle Tracking (LPT) has become a near-standard approach for performing accurate 3D flow measurements, thanks notably to the technical breakthroughs brought by the Iterative Particle Reconstruction (IPR: Wieneke, 2013) and Shake-the-Box (STB: Schanz et.al, 2016) procedures. These decisive progresses have triggered a number of studies relative to the eduction of flow kinematics and dynamics based on particle trajectory analyses. Novara & Scarano (2013), and others, focused on polynomial approximations of the trajectories, which analytically provide the material derivatives used to estimate pressure gradients. In particular, approximations based on second order polynomials fits of a small number of particle positions are used in commercially available softwares and among research teams as a straightforward solution to obtain the first and second order derivatives with a limited effect of the measurement noise. Additionally the analyses conducted during the 2020 LPT challenge (Leclaire, 2020 ; Sciacchitano, 2020) have addressed the performance of methodologies used by different groups with respect to second order trajectory fits for both multi-pulse and four-pulse (Novara et. al, 2016) LPT cases. On more advanced theoretical grounds, Geseman et. al (2016) have proposed the trackfit approach using penalized B-splines with considerations on the time-varying acceleration rate (i.e. jolt or jerk) and spectral content of noisy particle tracks

Runge–Kutta Pairs of Orders 6(5) with Coefficients Trained to Perform Best on Classical Orbits

We consider a family of explicit Runge–Kutta pairs of orders six and five without any additional property (reduced truncation errors, Hamiltonian preservation, symplecticness, etc.). This family offers five parameters that someone chooses freely. Then, we train them in order for the presented method to furnish the best results on a couple of Kepler orbits, a certain interval and tolerance. Consequently, we observe an efficient performance on a wide range of orbital problems (i.e., Kepler for a variety of eccentricities, perturbed Kepler with various disturbances, Arenstorf and Pleiades). About 1.8 digits of accuracy is gained on average over conventional pairs, which is truly remarkable for methods coming from the same family and order.

The planar two-body problem for spheroids and disks

We outline a new method suggested by Conway (CMDA 125:161–194, 2016) for solving the two-body problem for solid bodies of spheroidal or ellipsoidal shape. The method is based on integrating the gravitational potential of one body over the surface of the other body. When the gravitational potential can be analytically expressed (as for spheroids or ellipsoids), the gravitational force and mutual gravitational potential can be formulated as a surface integral instead of a volume integral and solved numerically. If the two bodies are infinitely thin disks, the surface integral has an analytical solution. The method is exact as the force and mutual potential appear in closed-form expressions, and does not involve series expansions with subsequent truncation errors. In order to test the method, we solve the equations of motion in an inertial frame and run simulations with two spheroids and two infinitely thin disks, restricted to torque-free planar motion. The resulting trajectories display precession patterns typical for non-Keplerian potentials. We follow the conservation of energy and orbital angular momentum and also investigate how the spheroid model approaches the two cases where the surface integral can be solved analytically, i.e., for point masses and infinitely thin disks.

Convergency and Stability of Explicit and Implicit Schemes in the Simulation of the Heat Equation

Some strategies for solving differential equations based on the finite difference method are presented: forward time centered space (FTSC), backward time centered space (BTSC), and the Crank-Nicolson scheme (CN). These are developed and applied to a simple problem involving the one-dimensional (1D) (one spatial and one temporal dimension) heat equation in a thin bar. The numerical implementation in this work can be used as a preamble to introduce a method of solving the heat equation that can be implemented in problems in the area of finances. The results of implementing the software on very fine meshes (unidimensional), and with relatively small-time steps, are shown. Through mesh refinement, it was possible to obtain a better temperature distribution in the thin bar between a range of points. The heat equation was solved numerically by testing both implicit (CN) and explicit (FTSC and BTSC) methods. The examples show that the implemented schemes conform to theoretical predictions and that truncation errors depend on mesh, spacing, and time step.

Quantifying and attributing time step sensitivities in present-day climate simulations conducted with EAMv1

This study assesses the relative importance of time integration error in present-day climate simulations conducted with the atmosphere component of the Energy Exascale Earth System Model version 1 (EAMv1) at 1∘ horizontal resolution. We show that a factor-of-6 reduction of time step size in all major parts of the model leads to significant changes in the long-term mean climate. Examples of changes in 10-year mean zonal averages include the following: up to 0.5 K of warming in the lower troposphere and cooling in the tropical and subtropical upper troposphere, 1 %–10 % decreases in relative humidity throughout the troposphere, and 10 %–20 % decreases in cloud fraction in the upper troposphere and decreases exceeding 20 % in the subtropical lower troposphere. In terms of the 10-year mean geographical distribution, systematic decreases of 20 %–50 % are seen in total cloud cover and cloud radiative effects in the subtropics. These changes imply that the reduction of temporal truncation errors leads to a notable although unsurprising degradation of agreement between the simulated and observed present-day climate; to regain optimal climate fidelity in the absence of those truncation errors, the model would require retuning. A coarse-grained attribution of the time step sensitivities is carried out by shortening time steps used in various components of EAM or by revising the numerical coupling between some processes. Our analysis leads to the finding that the marked decreases in the subtropical low-cloud fraction and total cloud radiative effect are caused not by the step size used for the collectively subcycled turbulence, shallow convection, and stratiform cloud macrophysics and microphysics parameterizations but rather by the step sizes used outside those subcycles. Further analysis suggests that the coupling frequency between the subcycles and the rest of EAM significantly affects the subtropical marine stratocumulus decks, while deep convection has significant impacts on trade cumulus. The step size of the cloud macrophysics and microphysics subcycle itself appears to have a primary impact on cloud fraction in the upper troposphere and also in the midlatitude near-surface layers. Impacts of step sizes used by the dynamical core and the radiation parameterization appear to be relatively small. These results provide useful clues for future studies aiming at understanding and addressing the root causes of sensitivities to time step sizes and process coupling frequencies in EAM. While this study focuses on EAMv1 and the conclusions are likely model-specific, the presented experimentation strategy has general value for weather and climate model development, as the methodology can help researchers identify and understand sources of time integration error in sophisticated multi-component models.

Effective Diahaline Diffusivities in Estuaries

<p>The present study aims to estimate effective diahaline turbulent salinity fluxes and diffusivities in numerical model simulations of estuarine scenarios. The underlying method is based on a quantification of salinity mixing per salinity class, which is shown to be twice the turbulent salinity transport across the respective isohaline. Using this relation, the recently derived universal law of estuarine mixing, predicting that average mixing per salinity class is twice the respective salinity times the river run‐off, can be directly derived. The turbulent salinity transport is accurately decomposed into physical (due to the turbulence closure) and numerical (due to truncation errors of the salinity advection scheme) contributions. The effective diahaline diffusivity representative for a salinity class and an estuarine region results as the ratio of the diahaline turbulent salinity transport and the respective (negative) salinity gradient, both integrated over the isohaline area in that region and averaged over a specified period. With this approach, the physical (or numerical) diffusivities are calculated as half of the product of physical (or numerical) mixing and the isohaline volume, divided by the square of the isohaline area. The method for accurately calculating physical and numerical diahaline diffusivities is tested and demonstrated for a three‐dimensional idealized exponential estuary. As a major product of this study, maps of the spatial distribution of the effective diahaline diffusivities are shown for the model estuary.</p>