A Class of A-Stable Order Four and Six Linear Multistep Methods for Stiff Initial Value Problems

A new three and five step block linear methods based on the Adams family for the direct solution of stiff initial value problems (IVPs) are proposed. The main methods together with the additional methods which constitute the block methods are derived via interpolation and collocation procedures. These methods are of uniform order four and six for the three and five step methods respectively. The stability analysis of the two methods indicates that the methods are A–stable, consistent and zero stable. Numerical results obtained using the proposed new block methods show that they are attractive for the solutions of stiff problems and compete favorably with the well-known Matlab stiff ODE solver ODE23S. Keywords: Linear multistep methods, initial value problems, interpolation and collocation

One Step Continuous Hybrid Block Method for the Solution of y'''=f(x,y,y',y'')

In this paper, we present a block method for the direct solution of third order initial value problems of ordinary differential equations. Collocation and interpolation approach was adopted to generate a continuous linear multistep method which was then solved for the independent solution to give a continuous block method. We evaluated the result at selected grid points to give a discrete block which eventually gave simultaneous solutions at both grid and off grid points.  The one-step block method is consistent and A -stable, with good region of absolute stability. Experimental results confirmed the superiority of the new scheme over an existing method. Keywords: consistent, convergent, collocation, hybrid points, independent solution, interpolation, zero stabl

Implicit Two Step Adam Moulton Hybrid Block Method with Two Off-Step Points for Solving Stiff Ordinary Differential Equations

A two step block hybrid Adam Moulton method of uniform order five is presented for the solution of stiff initial value problems. The individual schemes that made up the block method are obtained from the same continuous scheme which is applied to provide the solutions of stiff initial value problems on non overlapping intervals. The constructed block method is consistent, zero – stable and A – stable. Numerical results obtained using the new block method show that it is superior for stiff systems and competes well with existing ones. Keywords: stiff ODEs, Block Method, Adam Moulton method, Stabilit

The United States COVID-19 Forecast Hub dataset

Academic researchers, government agencies, industry groups, and individuals have produced forecasts at an unprecedented scale during the COVID-19 pandemic. To leverage these forecasts, the United States Centers for Disease Control and Prevention (CDC) partnered with an academic research lab at the University of Massachusetts Amherst to create the US COVID-19 Forecast Hub. Launched in April 2020, the Forecast Hub is a dataset with point and probabilistic forecasts of incident cases, incident hospitalizations, incident deaths, and cumulative deaths due to COVID-19 at county, state, and national, levels in the United States. Included forecasts represent a variety of modeling approaches, data sources, and assumptions regarding the spread of COVID-19. The goal of this dataset is to establish a standardized and comparable set of short-term forecasts from modeling teams. These data can be used to develop ensemble models, communicate forecasts to the public, create visualizations, compare models, and inform policies regarding COVID-19 mitigation. These open-source data are available via download from GitHub, through an online API, and through R packages

Single-Step Sixth Stage Implicit Runge-Kutta Method for First-Order Initial Value Problems of Ordinary Differential Equations

We present a single-step block hybrid method obtained through multistep collocation (MC) approach, by incorporating three off-grid interpolation and two off-grid collocation points in a continuous linear hybrid method as opposed to a similar work where two-step methods were obtained. The discrete hybrid methods are used in the formulation a block method which is then reformulated into a new single-step sixth-stage implicit Runge-Kutta method (SIRK) for the numerical treatment of first-order ordinary differential equations. The basic convergence and A-stability properties of the new method are established. Numerical experiments performed using the new method further revealed a reduction in the absolute errors, showing that this method is a better candidate for similar existing ones in the literature, as such it should be used for such class of problems