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The Use of Spectral Methods in Bidomain Studies
Published in Theo C. Pilkington, Bruce Loftis, Joe F. Thompson, Savio L-Y. Woo, Thomas C. Palmer, Thomas F. Budinger, High-Performance Computing in Biomedical Research, 2020
Natalia Trayanova, Theo Pilkington
Most often, boundary value problems such as the present one, involving determining the potential and current distribution in the region of interest, are solved numerically by means of the finite difference method. Besides finite differences, other methods for solving partial differential equations in boundary value problems involve finite element, Monte Carlo, spectral, and variational methods. Finite element methods are often preferred since they allow considerable freedom in allocating computational elements wherever needed, important when highly irregular geometries are involved. Spectral methods are preferred for very regular geometries and smooth functions; typically, they converge toward a correct solution more rapidly than finite differences methods.39 A class of elliptic differential equations with constant coefficients which involve even-order derivatives as well as the function itself is very efficiently solved on a domain of regular geometry when employing Fourier transforms (series or integrals). The differential equation is converted into a set of linear algebraic equations for the transform of the original function that can be solved directly (the set of equations has a “diagonal” form).40 The equations governing a bidomain that contain constant conductivities and membrane resistance (passive bidomain) belong to the above class of elliptic equations.
Parallel Spectral Computations of Complex Engineering Flows
Published in Hojjat Adeli, Supercomputing in Engineering Analysis, 2020
George Em Karniadakis, Steven A. Orszag
Spectral methods are a class of discretization techniques for partial differential equations in which the solution to the problem is reconstructed from a set of expansion coefficients and an appropriate basis, typically Fourier or polynomial series (Gottlieb and Orszag, 1977). The coefficients are determined by weighted-residual projection of the continuous equation upon a finite-dimensional subspace. For sufficiently smooth solutions, spectral methods yield exponential convergence so that highly accurate solutions can be obtained with fewer number of degrees of freedom than are typically used in standard difference techniques. The global character of spectral discretizations, however, responsible for this faster than algebraic convergence, makes the method considerably more complex computationally as the resulting discrete system matrices lack sparsity, while the total computational expense is typically higher than other discretization techniques for the same size problem.
Computer Modelling of Microwave Sources
Published in R A Cairns, A D R Phelps, P Osborne, Generation and Application of High Power Microwaves, 2020
Spectral methods correspond to Galerkin finite element methods with global orthogonal functions (Fourier harmonics or Chebychev polynomials) being used instead of local polynomials. Pseudospectral methods correspond to point collocation finite element methods. For linear terms, spectral and pseudo-spectral give the same results. Quadratic terms differ in that the pseudo-spectral method has periodic conditions in wavenumber space. For poorly resolved spectra, this can lead to nonlinear instabilities (see section 5.2). A ‘dealiased’ pseudo-spectral scheme, where the top third of the spectrum is set to zero when using finite fourier transforms, is mathematically identical to a spectral approximation.
Solving nonlinear diffusive problems in buildings by means of a Spectral reduced-order model
Published in Journal of Building Performance Simulation, 2019
Suelen Gasparin, Julien Berger, Denys Dutykh, Nathan Mendes
While finite-difference and finite-element methods are based on a local representation of functions, using low-order approximations, Spectral methods consider a global representation of the solution, which yields beyond all orders approximations (Boyd 2000). In the global representation approach, the value of the derivative at a certain spatial location depends on the solution on the entire domain and not only on its neighbours. Spectral methods consider a sum of polynomials that suit for this whole domain, almost like an analytical solution, providing a high approximation of the solution. As its error decreases exponentially, it is possible to have the same accuracy of other methods but with a lower number of modes, which makes this method memory usage minimized, allowing to store and operate a lower number of degrees of freedom (Trefethen 1996). The Spectral methods used in this work are the Chebyshev polynomials on the basis function and the Tau–Galerkin method to compute the temporal coefficients.
Space–time spectral collocation method for one-dimensional PDE constrained optimisation
Published in International Journal of Control, 2020
A. Rezazadeh, M. Mahmoudi, M. Darehmiraki
This paper solves the parabolic constrained optimal control problem with space–time spectral collocation method for discretising spatial and time derivatives, which is accurate to a high degree, in both space and time. Spectral methods have been used successfully to solve elliptic partial differential equations for many years. If the exact solution is analytic, the numerical solution will converge exponentially quickly as a function which is dependant on the number of spectral modes. Spectral methods generally provide the most accurate solutions which are sufficiently smooth and also effective for nonlinear problems.
Impact of Joule heating and nonlinear thermal radiation on the flow of Casson nanofluid with entropy generation
Published in International Journal of Ambient Energy, 2022
K. Sruthila Gopalakrishnan, Ibukun Sarah Oyelakin, Sabyasachi Mondal, Ram Prakash Sharma
Spectral methods are among the most successful numerical techniques used to solve differential equations with applications in fluid dynamics, mathematical biology, material sciences and quantum physics. In the spectral method, a function is approximated globally by applying interpolating polynomials of higher order. Spectral methods have been widely used to solve many research work concerned with nanofluids. Kameswaran, Sibanda, and Motsa (2013) used the spectral relaxation method (SRM) to analyse the influence of thermal dispersion and radiation on the unsteady flow of three distinct nanofluids, which are water-based with the nanoparticles: titanium oxide (TiO2), copper oxide (CuO) and aluminium oxide (Al2O3), past a stretching sheet. SRM proved to be an effective alternative to Keller box and Runge–Kutta methods in solving boundary value problems. Haroun, Mondal, and Sibanda (2015a, 2015b) examined heat and mass transfer in the unsteady magnetohydrodynamic nanofluid flow caused by a stretching surface. Haroun, Mondal, and Sibanda (2015a, 2015b) used the spectral method to study a revised model of Haroun, Mondal, and Sibanda (2015a, 2015b). In the study, they considered the impact of Brownian motion and thermophoresis diffusion on the boundary flow of MHD nanofluid over an extended surface, with the volume fraction flux of nanoparticles at the boundary being passively controlled. This, in turn, seems to be a problem that is very much related to the real world. Agbaje et al. (2018) introduced a new numerical approach, the spectral perturbation method (SPM). They used the technique to analyse near the stagnation point flow and heat transfer in a fluid, which resists compression and conducts electricity. Gangadhar, Bhargavi, and Munagala (2020a, 2020b) examined heat transfer in a laminar boundary layer flow of Casson fluid with viscous dissipation. They showed that an increase in the Casson parameter could condense the rate of heat transfer show opposite effects in the coefficient of skin friction. Oyelakin et al. (2020) with the application of spectral local linearisation method (SLLM) analysed the effect of double diffusion convection upon the 3-D MHD flow of tangent hyperbolic Casson nanofluid. Shahid et al. (2021) with the utilisation of spectral local linearisation method (SSLM) analysed the MHD flow of nanofluid upon a permeable sheet under the influence of viscosity which depends on the temperature exponentially. Verma and Mondal (2021) presented a brief review on different numerical methods which can be utilised to study the transmission of heat as well as mass in Casson fluids. It was noted that among various numerical methods, spectral methods are more efficient in solving the problems concerned with Casson fluid. Rai and Mondal (2021) provided a review of spectral methods utilised to solve various types of flow of fluids, especially flow of nanofluids.