Spectrally accurate space-time solution of Hamiltonian PDEs

Recently, the numerical solution of multi-frequency, highly-oscillatory Hamiltonian problems has been attacked by using Hamiltonian Boundary Value Methods (HBVMs) as spectral methods in time. When the problem derives from the space semi- discretization of (possibly Hamiltonian) partial differential equations (PDEs), the resulting problem may be stiffly-oscillatory, rather than highly-oscillatory. In such a case, a different implementation of the methods is needed, in order to gain the maximum efficiency.Comment: 17 pages, 3 figure

Singly TASE Operators for the Numerical Solution of Stiff Differential Equations by Explicit Runge–Kutta Schemes

In this paper new explicit integrators for numerical solution of stiff evolution equations are proposed. As shown by Bassenne, Fu and Mani in (J Comput Phys 424:109847, 2021), the action on the original vector field of the stiff equations of an appropriate time-accurate and highly-stable explicit (TASE) linear operator, allows us to use explicit Runge–Kutta (RK) schemes with these modified equations so that the resulting algorithm becomes stable for the original stiff equations. Here a new family of TASE operators is considered. The new operators, called Singly TASE, have the advantage over the TASE operators of Bassenne et al. that the action on the vector field depends on the powers of the inverse of only one matrix, which can be computationally more simple, without loosing stability properties. A complete study of the linear stability properties of k–stage, kth–order explicit RK schemes under the action of Singly TASE operators of the same order is carried out for k≤4. For orders two, three and four, particular schemes that are nearly strongly A–stable and therefore suitable for stiff problems are devised. Further, explicit RK schemes with orders three and four that can be implemented with only two storage locations under the action of Singly TASE operators of the same order are discussed. A particular implementation of the classical four–stage fourth–order RK scheme with two Singly TASE operators is presented. A set of numerical experiments has been conducted to demonstrate the performance of the new schemes by comparing with previous RKTASE and other established methods. The main conclusion is that the new integrators provide a very simple solver for stiff systems with good stability properties and avoids the difficulties of using implicit algorithms

Spatio temporal dynamics of direct current in treated anisotropic tumors

The inclusion of a diffusion term in the modified Gompertz equation (Cabrales et al., 2018) allows to describe the spatiotemporal growth of direct current treated tumors. The aim of this study is to extend the previous model to the case of anisotropic tumors, simulating the spatiotemporal behavior of direct current treated anisotropic tumors, also carrying out a theoretical analysis of the proposed model. Growths in the mass, volume and density of the solid tumors are shown for each response type after direct current application (disease progression, partial response, stationary partial response and complete remission). For this purpose, the Method of Lines and different diffusion tensors are used. The results show that the growth of the tumor treated with direct current is faster for the shorter duration of the net antitumor effect and the higher diffusion coefficient and anisotropy degree of the solid tumor. It is concluded that the greatest direct current antitumor effectiveness occurs for the highly heterogeneous, anisotropic, aggressive and hypodense malignant solid tumors

Modified SEIR epidemic model including asymptomatic and hospitalized cases with correct demographic evolution

The aim of this study is to propose a modified Susceptible-Exposed-Infectious-Removed (SEIR) model that describes the time behaviour of symptomatic, asymptomatic and hospitalized patients in an epidemic, taking into account the effect of the demographic evolution. Unlike most of the recent studies where a constant ratio of new individuals is considered, we consider a more correct assumption that the growth ratio is proportional to the total population, following a Logistic law, as is usual in population growth studies for humans and animals. An exhaustive theoretical study is carried out and the basic reproduction number is computed from the model equations. It is proved that if then the disease-free manifold is globally asymptotically stable, that is, the epidemics remits. Global and local stability of the equilibrium points is also studied. Numerical simulations are used to show the agreement between numerical results and theoretical properties. The model is fitted to experimental data corresponding to the pandemic evolution of COVID-19 in the Republic of Cuba, showing a proper behaviour of infected cases which let us think that can provide a correct estimation of asymptomatic cases. In conclusion, the model seems to be an adequate tool for the study and control of infectious diseases

Fast distributed consensus with Chebyshev polynomials

Abstract — Global observation of the environment is a key component in sensor networks and multi-robot systems. Dis-tributed consensus algorithms make all the nodes in the network to achieve a common perception by local interactions between direct neighbors. The convergence rate of these algorithms depends on the network connectivity, which is related to the second largest eigenvalue of the weighted adjacency matrix of the communication graph. When the connectivity is small, a large number of communication rounds is required to achieve the consensus. In this paper we present a new distributed consensus algorithm which uses the properties of Chebyshev polynomials to significantly increase the convergence rate. The algorithm is expressed in the form of a linear iteration and, at each step, the nodes only require to transmit their current state to their neighbors. The difference with respect to previous approaches is that our algorithm is based on a second order difference equation. We provide the analytical expression of the convergence rate and we study in which conditions it is faster than computing the powers of the weighted matrix. This improvement reduces the number of messages between nodes, saving both power and time to the networked system. We evaluate our algorithm in a simulated environment showing the benefits of our approach

Estudio de técnicas de discretización en problemas de consenso distribuido

En este Tfg, partiendo de una red de sensores que nos definen un grafo se ha estudiado el problema de consenso, a través de las ecuaciones diferenciales que lo definen. Como se trata de un problema distribuido, no se puede resolver de manera analítica. Por lo tanto, se ha intentado buscar un método de discretización alternativo al de Euler, que es la utilizada en este tipo de problemas. Dicho método, presenta algunos inconvenientes tiempo entre dos comunicaciones sucesivas. Para solucionar este problema, se ha estudiado los métodos multi-paso. En nuestro caso, se ha utilizado un método de dos pasos y orden uno y se han obsevado las propiedades que presenta dicho métod

Explicit Runge-Kutta methods for stiff problems with a gap in their eigenvalue spectrum

In this paper we consider the numerical solution of stiff problems in which the eigenvalues are separated into two clusters, one containing the "stiff", or fast, components and one containing the slow components, that is, there is a gap in their eigenvalue spectrum. By using exponential fitting techniques we develop a class of explicit Runge-Kutta methods, that we call stability fitted methods, for which the stability domain has two regions, one close to the origin and the other one fitting the large eigenvalues. We obtain the size of their stability regions as a function of the order and the fitting conditions. We also obtain conditions that the coefficients of these methods must satisfy to have a given stiff order for the Prothero-Robinson test equation. Finally, we construct an embedded pair of stability fitted methods of orders 2 and 1 and show its performance by means of several numerical experiments

Approximate Compositions of a Near Identity Map by Multi-Revolution Runge-Kutta Methods

The so-called multi-revolution methods were introduced in celestial mechanics as an e#cient tool for the long-term numerical integration of nearly periodic orbits of artificial satellites around the Earth. A multi-revolution method is an algorithm that approximates the map # T of N near-periods T in terms of the one near-period map #T evaluated at few s N selected points. More generally, multi-revolution methods aim at approximating the composition # of a near identity map #. In this paper we give a general presentation and analysis of multi-revolution Runge-Kutta (MRRK) methods similar to the one developed by Butcher for standard Runge-Kutta methods applied to ordinary di#erential equations. Order conditions, simplifying assumptions, and order estimates of MRRK methods are given. MRRK methods preserving constant Poisson/symplectic structures and reversibility properties are characterized. The construction of high order MRRK methods is described based on some families of orthogonal polynomials