Pseudospectral Roaming Contour Integral Methods for Convection-Diffusion Equations

AbstractWe generalize ideas in the recent literature and develop new ones in order to propose a general class of contour integral methods for linear convection–diffusion PDEs and in particular for those arising in finance. These methods aim to provide a numerical approximation of the solution by computing its inverse Laplace transform. The choice of the integration contour is determined by the computation of a few suitably weighted pseudo-spectral level sets of the leading operator of the equation. Parabolic and hyperbolic profiles proposed in the literature are investigated and compared to the elliptic contour originally proposed by Guglielmi, López-Fernández and Nino 2020, see Guglielmi et al. (Math Comput 89:1161–1191, 2020). In summary, the article provides a comparison among three different integration profiles; proposes a new fast pseudospectral roaming method; optimizes the selection of time windows on which one may arbitrarily approximate the solution by no extra computational cost with respect to the case of a fixed time instant; focuses extensively on computational aspects and it is the reference of the MATLAB code [20], where all algorithms described here are implemented.

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Approximation of linear one dimensional partial differential equations including fractional derivative with non-singular kernel

Advances in Difference Equations ◽

10.1186/s13662-021-03472-z ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Raheel Kamal ◽

Kamran ◽

Gul Rahmat ◽

Ali Ahmadian ◽

Noreen Izza Arshad ◽

...

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Fractional Derivative ◽

Meshless Method ◽

Inverse Laplace Transform ◽

Contour Integral ◽

Partial Differential ◽

One Dimensional ◽

The Laplace Transform

AbstractIn this article we propose a hybrid method based on a local meshless method and the Laplace transform for approximating the solution of linear one dimensional partial differential equations in the sense of the Caputo–Fabrizio fractional derivative. In our numerical scheme the Laplace transform is used to avoid the time stepping procedure, and the local meshless method is used to produce sparse differentiation matrices and avoid the ill conditioning issues resulting in global meshless methods. Our numerical method comprises three steps. In the first step we transform the given equation to an equivalent time independent equation. Secondly the reduced equation is solved via a local meshless method. Finally, the solution of the original equation is obtained via the inverse Laplace transform by representing it as a contour integral in the complex left half plane. The contour integral is then approximated using the trapezoidal rule. The stability and convergence of the method are discussed. The efficiency, efficacy, and accuracy of the proposed method are assessed using four different problems. Numerical approximations of these problems are obtained and validated against exact solutions. The obtained results show that the proposed method can solve such types of problems efficiently.

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Determination of the order of fractional derivative for subdiffusion equations

Fractional Calculus and Applied Analysis ◽

10.1515/fca-2020-0081 ◽

2020 ◽

Vol 23 (6) ◽

pp. 1647-1662

Author(s):

Ravshan Ashurov ◽

Sabir Umarov

Keyword(s):

Fractional Derivative ◽

Fixed Time ◽

Time Instant ◽

Observation Data ◽

Additional Information ◽

Fractional Modeling ◽

Time Fractional Derivative ◽

The Right ◽

Right Order

Abstract The identification of the right order of the equation in applied fractional modeling plays an important role. In this paper we consider an inverse problem for determining the order of time fractional derivative in a subdiffusion equation with an arbitrary second order elliptic differential operator. We prove that the additional information about the solution at a fixed time instant at a monitoring location, as “the observation data”, identifies uniquely the order of the fractional derivative.

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Microscale dynamics of electrophysiological markers of epilepsy

10.1101/2020.10.14.20211649 ◽

2020 ◽

Author(s):

Jimmy C. Yang ◽

Angelique C. Paulk ◽

Sang Heon Lee ◽

Mehran Ganji ◽

Daniel J. Soper ◽

...

Keyword(s):

Spatial Resolution ◽

High Frequency ◽

Time Windows ◽

List Type ◽

Spatiotemporal Resolution ◽

Similar Time ◽

Human Epilepsy ◽

High Frequency Oscillations ◽

Peak Tracking ◽

Interictal Discharges

AbstractObjectiveInterictal discharges (IIDs) and high frequency oscillations (HFOs) are neurophysiologic biomarkers of epilepsy. In this study, we use custom poly(3,4-ethylenedioxythiophene) polystyrene sulfonate (PEDOT:PSS) microelectrodes to better understand their microscale dynamics.MethodsElectrodes with spatial resolution down to 50µm were used to record intraoperatively in 30 subjects. For IIDs, putative spatiotemporal paths were generated by peak-tracking, followed by clustering. For HFOs, repeating patterns were elucidated by clustering similar time windows. Fast events, consistent with multi-unit activity (MUA), were covaried with either IIDs or HFOs.ResultsIIDs seen across the entire array were detected in 93% of subjects. Local IIDs, observed across <50% of the array, were seen in 53% of subjects. IIDs appeared to travel across the array in specific paths, and HFOs appeared in similar repeated spatial patterns. Finally, microseizure events were identified spanning 50-100µm. HFOs covaried with MUA, but not with IIDs.ConclusionsOverall, these data suggest micro-domains of irritable cortex that form part of an underlying pathologic architecture that contributes to the seizure network.SignificanceMicroelectrodes in cases of human epilepsy can reveal dynamics that are not seen by conventional electrocorticography and point to new possibilities for their use in the diagnosis and treatment of epilepsy.HighlightsPEDOT:PSS microelectrodes with at least 50µm spatial resolution uniquely reveal spatiotemporal patterns of markers of epilepsyHigh spatiotemporal resolution allows interictal discharges to be tracked and reveal cortical domains involved in microseizuresHigh frequency oscillations detected by microelectrodes demonstrate localized clustering on the cortical surface

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Hybrid Nodal Integral / Finite Element Method for Time-Dependent Convection Diffusion Equation

Journal of Nuclear Engineering and Radiation Science ◽

10.1115/1.4051928 ◽

2021 ◽

Author(s):

Sundar Namala ◽

Rizwan Uddin

Keyword(s):

Finite Element Method ◽

Finite Element ◽

Diffusion Equation ◽

Numerical Solutions ◽

Second Order ◽

Time Dependent ◽

Convection Diffusion ◽

Convection Diffusion Equation ◽

Integral Methods ◽

Element Method

Abstract Nodal integral methods (NIM) are a class of efficient coarse mesh methods that use transverse averaging to reduce the governing partial differential equation(s) (PDE) into a set of ordinary differential equations (ODE). The standard application of NIM is restricted to domains that have boundaries parallel to one of the coordinate axes/palnes (in 2D/3D). The hybrid nodal-integral/finite-element method (NI-FEM) reported here has been developed to extend the application of NIM to arbitrary domains. NI-FEM is based on the idea that the interior region and the regions with boundaries parallel to the coordinate axes (2D) or coordinate planes (3D) can be solved using NIM, and the rest of the domain can be discretized and solved using FEM. The crux of the hybrid NI-FEM is in developing interfacial conditions at the common interfaces between the NIM regions and FEM regions. We here report the development of hybrid NI-FEM for the time-dependent convection-diffusion equation (CDE) in arbitrary domains. Resulting hybrid numerical scheme is implemented in a parallel framework in Fortran and solved using PETSc. The preliminary approach to domain decomposition is also discussed. Numerical solutions are compared with exact solutions, and the scheme is shown to be second order accurate in both space and time. The order of approximations used for the development of the scheme are also shown to be second order. The hybrid method is more efficient compared to standalone conventional numerical schemes like FEM.

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A Sequential Approach to Numerical Simulations of Solidification with Domain and Time Decomposition

Applied Sciences ◽

10.3390/app9101972 ◽

2019 ◽

Vol 9 (10) ◽

pp. 1972 ◽

Cited By ~ 5

Author(s):

Elzbieta Gawronska

Keyword(s):

Parallel Computing ◽

Computational Cost ◽

Fixed Time ◽

Simulation Software ◽

Computational Method ◽

Computational Time ◽

Time Step ◽

Sequential Approach ◽

Time Partitioning ◽

Physical Phenomena

Progress in computational methods has been stimulated by the widespread availability of cheap computational power leading to the improved precision and efficiency of simulation software. Simulation tools become indispensable tools for engineers who are interested in attacking increasingly larger problems or are interested in searching larger phase space of process and system variables to find the optimal design. In this paper, we show and introduce a new approach to a computational method that involves mixed time stepping scheme and allows to decrease computational cost. Implementation of our algorithm does not require a parallel computing environment. Our strategy splits domains of a dynamically changing physical phenomena and allows to adjust the numerical model to various sub-domains. We are the first (to our best knowledge) to show that it is possible to use a mixed time partitioning method with various combination of schemes during binary alloys solidification. In particular, we use a fixed time step in one domain, and look for much larger time steps in other domains, while maintaining high accuracy. Our method is independent of a number of domains considered, comparing to traditional methods where only two domains were considered. Mixed time partitioning methods are of high importance here, because of natural separation of domain types. Typically all important physical phenomena occur in the casting and are of high computational cost, while in the mold domains less dynamic processes are observed and consequently larger time step can be chosen. Finally, we performed series of numerical experiments and demonstrate that our approach allows reducing computational time by more than three times without losing the significant precision of results and without parallel computing.

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A Simple Time-Varying Sensitivity Analysis (TVSA) for Assessment of Temporal Variability of Hydrological Processes

Water ◽

10.3390/w12092463 ◽

2020 ◽

Vol 12 (9) ◽

pp. 2463 ◽

Cited By ~ 1

Author(s):

Yelena Medina ◽

Enrique Muñoz

Keyword(s):

Sensitivity Analysis ◽

Temporal Variability ◽

Time Windows ◽

Computational Cost ◽

Time Varying ◽

Moving Window ◽

Time Sensitivity ◽

Annual Time ◽

Low Computational Cost ◽

High Computational Cost

Time-varying sensitivity analysis (TVSA) allows sensitivity in a moving window to be estimated and the time periods in which the specific components of a model can affect its performance to be identified. However, one of the disadvantages of TVSA is its high computational cost, as it estimates sensitivity in a moving window within an analyzed series, performing a series of repetitive calculations. In this article a function to implement a simple TVSA with a low computational cost using regional sensitivity analysis is presented. As an example of its application, an analysis of hydrological model results in daily, monthly, and annual time windows is carried out. The results show that the model allows the time sensitivity of a model with respect to its parameters to be detected, making it a suitable tool for the assessment of temporal variability of processes in models that include time series analysis. In addition, it is observed that the size of the moving window can influence the estimated sensitivity; therefore, analysis of different time windows is recommended.

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Computation of Pressure Fields around a Two-Dimensional Circular Cylinder Using the Vortex-In-Cell and Penalization Methods

Modelling and Simulation in Engineering ◽

10.1155/2014/708372 ◽

2014 ◽

Vol 2014 ◽

pp. 1-13 ◽

Cited By ~ 4

Author(s):

Seung-Jae Lee ◽

Jun-Hyeok Lee ◽

Jung-Chun Suh

Keyword(s):

Circular Cylinder ◽

Solid Body ◽

Pressure Field ◽

Stokes Equations ◽

Computational Cost ◽

Fixed Time ◽

Computational Grids ◽

Penalty Term ◽

Pressure Fields ◽

Wide Range

The vorticity-velocity formulation of the Navier-Stokes equations allows purely kinematical problems to be decoupled from the pressure term, since the pressure is eliminated by applying the curl operator. The Vortex-In-Cell (VIC) method, which is based on the vorticity-velocity formulation, offers particle-mesh algorithms to numerically simulate flows past a solid body. The penalization method is used to enforce boundary conditions at a body surface with a decoupling between body boundaries and computational grids. Its main advantage is a highly efficient implementation for solid boundaries of arbitrary complexity on Cartesian grids. We present an efficient algorithm to numerically implement the vorticity-velocity-pressure formulation including a penalty term to simulate the pressure fields around a solid body. In vorticity-based methods, pressure field can be independently computed from the solution procedure for vorticity. This clearly simplifies the implementation and reduces the computational cost. Obtaining the pressure field at any fixed time represents the most challenging goal of this study. We validate the implementation by numerical simulations of an incompressible viscous flow around an impulsively started circular cylinder in a wide range of Reynolds numbers: Re=40, 550, 3000, and 9500.

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Looking Forward Approach in Cooperative Differential Games

International Game Theory Review ◽

10.1142/s0219198916400077 ◽

2016 ◽

Vol 18 (02) ◽

pp. 1640007 ◽

Cited By ~ 5

Author(s):

Petrosian Ovanes

Keyword(s):

Differential Games ◽

Fixed Time ◽

Time Instant ◽

Time Interval ◽

New Approach ◽

Game Dynamics ◽

Motion Equations ◽

Payoff Functions ◽

Definition Of ◽

Cooperative Differential Games

New approach to the definition of solution in cooperative differential games is considered. The approach is based on artificially truncated information about the game. It assumed that at each time, instant players have information about the structure of the game (payoff functions, motion equations) only for the next fixed time interval. Based on this information they make the decision. Looking Forward Approach is applied to the cases when the players are not sure about the dynamics of the game on the whole time interval [Formula: see text] and orient themselves on the game dynamics defined on the smaller time interval [Formula: see text] ([Formula: see text]), on which they surely know that the game dynamics is not changing.

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