Showing papers on "Tridiagonal matrix algorithm published in 2020"
TL;DR: In this paper, a two-directional unconditionally stable single field (US-SF) finite-difference time-domain (FDTD) method is generalized to a 3D vector wave equation.
Abstract: The recently developed two-directional unconditionally stable single-field (US-SF) finite-difference time-domain (FDTD) method is generalized to a 3-D. The method is based on the application of the Crank–Nicolson scheme to only one of Maxwell curl equations which leads to an unconditionally stable finite-difference solution of the 3-D vector wave equation in the time domain. The method is designated as a single field because only the electric (or magnetic) field is updated in each time step. The implicit equations for each time step can be solved using a tridiagonal matrix algorithm without a need for matrix inversion. To achieve this, we introduced a new modified time-splitting scheme for the multilevel difference equations. Unlike the existing time-splitting method used to solve such equations which results in instability when dealing with inhomogeneous media, the presented method remains stable. As an important feature of the proposed US-SF-FDTD method, the updating of the three field components can be executed simultaneously (in parallel) by applying multithreading, thereby significantly reducing runtime. The unconditional stability of the proposed method is proved analytically. The accuracy and computational efficiency of the proposed method are demonstrated by providing numerical examples and by comparison to other FDTD methods and to the analytic solution when available.
10 citations
TL;DR: It is shown that instabilities also occur when the coefficients of a regional Gaussian filter are calculated using a conventional recurrence relation, and methods for their proper treatment are provided.
9 citations
6 citations
4 Aug 2020
TL;DR: In this paper, the authors presented a time scale dimensionless model that considers charge carrier motion and ion migration in a perovskite solar cell, which provides high accuracy accompanied by the use of realistic parameters.
Abstract: An important tool for explaining the hysteretic behavior in movement of electronic and ionic charges is drift-diffusion model. Adding numerical methods to these models in realistic operation situations is challenging due to the fact that some parameters have extreme values. We present a time scale dimensionless model that considers charge carrier motion and ion migration in a perovskite solar cell. The proposed model provides high accuracy accompanied by the use of realistic parameters. In order to solve matrix and equations, tridiagonal matrix algorithm (TDMA) method is exploited. Electric potential, density of ion vacancy migration, hole and electron concentration characteristics are calculated and illustrated in transient time scale. Besides, the mentioned characteristics are illustrated with different feasible built-in potential. This approach gives insight into device physics, charge transport model, ion migration and hysteresis phenomena.
5 citations
TL;DR: The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic- reduction (PCR) and lid-driven cavity problem is considered to assess these parallel approaches.
Abstract: The solution of tridiagonal system of equations using graphic processing units (GPU) is assessed. The parallel-Thomas-algorithm (PTA) is developed and the solution of PTA is compared to two known parallel algorithms, i.e. cyclic-reduction (CR) and parallel-cyclic-reduction (PCR). Lid-driven cavity problem is considered to assess these parallel approaches. This problem is also simulated using the classic Thomas algorithm that runs on a central processing unit (CPU). Runtimes and physical parameters of the mentioned GPU and CPU algorithms are compared. The results show that the speedup of CR, PCR and PTA against the CPU runtime is 4.4x ,5.2x and 38.5x , respectively. Furthermore, the effect of coalesced and uncoalesced memory access to GPU global memory is examined for PTA, and a 2x -speedup is achieved for the coalesced memory access. Additionally, the PTA performance in a time dependent problem, the unsteady flow over a square, is assessed and a 9x-speedup is obtained against the CPU.
4 citations
TL;DR: A recursive version of the previously proposed fast algorithm for computation of the field scattered from arbitrary-shaped multilayer objects that has lower time and memory complexity and supersedes the original version.
Abstract: We present a recursive version of the previously proposed fast algorithm for computation of the field scattered from arbitrary-shaped multilayer objects. As in the original version, the field at each layer is represented by a series of cylindrical functions with unknown coefficients. In order to determine these coefficients, a linear system of equations is obtained through a procedure based on the boundary conditions between the layers. Instead of solving this large system by applying conventional linear solvers as done in the original version, through an approach based on Thomas algorithm, we derive recursive expressions to compute the scattered field. While the accuracy of the results for both versions are almost the same, the recursive version has lower time and memory complexity. Hence, it supersedes the original version.
3 citations
TL;DR: An exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations, are proposed and the modified tridiagonal matrix algorithm is proposed to solve systems oflinear equations.
Abstract: We propose an exact three-point scheme and schemes of high order of accuracy, which are two systems of linear algebraic equations. Each equation of the system contains five unknown values of the exact solution and its first derivative at three grid points on the interval. In constructing the scheme, the principle of superposition of solutions was used. Partial sums of the functional series representing independent solutions provide schemes of arbitrary order of accuracy for the boundary-value poblem and for the spectral one. To solve systems of linear equations, the modified tridiagonal matrix algorithm is proposed.
2 citations
TL;DR: A breakdown-free symbolic algorithm for computing the inverse of an n -by- n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm is presented.
Abstract: Matrix inverse computation is one of the fundamental mathematical problems of linear algebra and has been widely used in many fields of science and engineering. In this paper, we consider the inverse computation of an opposite-bordered tridiagonal matrix which has attracted much attention in recent years. By exploiting the low-rank structure of the matrix, first we show that an explicit formula for the inverse of the opposite-bordered tridiagonal matrix can be obtained based on the combination of a specific matrix splitting and the generalized Sherman–Morrison–Woodbury formula. Accordingly, a numerical algorithm is outlined. Second, we present a breakdown-free symbolic algorithm of
$$O(n^2)$$
for computing the inverse of an n-by-n opposite-bordered tridiagonal matrix, which is based on the use of GTINV algorithm and the generalized symbolic Thomas algorithm.
Finally, we have provided the results of some numerical experiments for the sake of illustration.
1 citations
28 Jul 2020
TL;DR: In this article, the authors presented an exponentially fitted spline method to solve SPDDE with dual layer, where the given second order differential-difference equation is replaced by an asymptotically proportionate second order singular perturbation problem.
Abstract: In this paper, we presented exponentially fitted spline method to solve SPDDE with dual layer. At first, the given second order differential-difference equation is replaced by an asymptotically proportionate second order singular perturbation problem. At that point, a fitting factor is brought into the exponentially fitted spline Method. The value of fitting factor is obtained by the singular perturbations theory. The Thomas algorithm is used to solve the tridiagonal system obtained by the method. The result of the delay and also advance parameters on the boundary layer(s) has likewise been evaluated as well as represented in charts. The applicability of the proposed plan is actually confirmed through executing it on model examples. To show the accuracy of the method, the results are presented in terms of maximum absolute errors for arbitrary λ1, λ2 such that λ1 +λ2 =1/2.
1 citations
TL;DR: It turns out that a combination of the Thomas algorithm and the approximate inverse leads to a solution that does not need either tiling or transpositions, and none of the kernels uses an extensive amount of shared memory which yields a very high GPU utilization and more importantly optimal coalesced global memory access patterns.
12 Jul 2020
TL;DR: The main algorithm is a computing the electric3 algorithm for numerical-numerical-coding-of-electrical-modelling-augmented-reality problems to solve the current preconditioned problems, whereas the main computed algorithm is saturated by the saturated field, so the main algorithm allows the current problems to be solved.
Abstract: We present a numerical algorithm for computing the electric field in digital rock samples and estimating their electrical resistivity (conductivity). The main peculiarity of the algorithm is its applicability tostrongly heterogeneous models including partially saturated and multi-mineral rock samples. The algorithm is based on the iterative Krylov-type solver preconditioned by the inverse Laplace operator for homogeneous media. The preconditioner is computed using the spectral method in directions orthogonal to the direction of the main electric current, whereas the series of 1D problems are solved by the Thomas algorithm. We implement the algorithm using GPUs, which allows us to use a single GPU to solve the problems for samples whose size is up to 4003 voxels.