3rd and 11th orders of accuracy of ‘linear’ and ‘quadratic’ elements for Poisson equation with irregular interfaces on Cartesian meshes

Purpose The purpose of this paper is as follows: to significantly reduce the computation time (by a factor of 1,000 and more) compared to known numerical techniques for real-world problems with complex interfaces; and to simplify the solution by using trivial unfitted Cartesian meshes (no need in complicated mesh generators for complex geometry). Design/methodology/approach This study extends the recently developed optimal local truncation error method (OLTEM) for the Poisson equation with constant coefficients to a much more general case of discontinuous coefficients that can be applied to domains with different material properties (e.g. different inclusions, multi-material structural components, etc.). This study develops OLTEM using compact 9-point and 25-point stencils that are similar to those for linear and quadratic finite elements. In contrast to finite elements and other known numerical techniques for interface problems with conformed and unfitted meshes, OLTEM with 9-point and 25-point stencils and unfitted Cartesian meshes provides the 3-rd and 11-th order of accuracy for irregular interfaces, respectively; i.e. a huge increase in accuracy by eight orders for the new 'quadratic' elements compared to known techniques at similar computational costs. There are no unknowns on interfaces between different materials; the structure of the global discrete system is the same for homogeneous and heterogeneous materials (the difference in the values of the stencil coefficients). The calculation of the unknown stencil coefficients is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy of OLTEM at a given stencil width. The numerical results with irregular interfaces show that at the same number of degrees of freedom, OLTEM with the 9-points stencils is even more accurate than the 4-th order finite elements; OLTEM with the 25-points stencils is much more accurate than the 7-th order finite elements with much wider stencils and conformed meshes. Findings The significant increase in accuracy for OLTEM by one order for 'linear' elements and by 8 orders for 'quadratic' elements compared to that for known techniques. This will lead to a huge reduction in the computation time for the problems with complex irregular interfaces. The use of trivial unfitted Cartesian meshes significantly simplifies the solution and reduces the time for the data preparation (no need in complicated mesh generators for complex geometry). Originality/value It has been never seen in the literature such a huge increase in accuracy for the proposed technique compared to existing methods. Due to a high accuracy, the proposed technique will allow the direct solution of multiscale problems without the scale separation.

An efficient method for oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics

In this paper, a novel numerical technique, the first-order Smooth Composite Chebyshev Finite Difference method, is presented. Imposing a first-order smoothness of the approximation polynomial at the ends of each subinterval is originality of the method. Both round-off and truncation error analyses of the method are performed beside the convergence analysis. Diffusion of oxygen in a spherical cell including nonlinear uptake kinetics is solved by using the method. The obtained results are compared with the existing methods in the literature and it is observed that the proposed method gives more reliable results.

A novel second-order adaptive grid method for singularly perturbed convection–diffusion equations

In this paper, a singularly perturbed convection–diffusion equation is studied. At first, the original problem is transformed into a parameterized singularly perturbed Volterra integro-differential equation by using an integral transform. Then, a second-order finite difference method on an arbitrary mesh is given. The stability and local truncation error estimates of the discrete schemes are analyzed. Based on the mesh equidistribution principle and local truncation error estimation, an adaptive grid algorithm is given. In addition, in order to calculate the parameters of the transformation equation, a nonlinear unconstrained optimization problem is constructed. Numerical experiments are given to illustrate the effectiveness of our presented adaptive grid algorithm.

Deep Learning Neural Network Algorithm for Computation of SPICE Transient Simulation of Nonlinear Time Dependent Circuits

In this paper, a special method based on the neural network is presented, which is conveniently used to precompute the steps of numerical integration. This method approximates the behaviour of the numerical integrator with respect to the local truncation error. In other words, it allows the precomputation of the individual steps in such a way that they do not need to be estimated by an algorithm but can be directly estimated by a neural network. Experimental tests were performed on a series of electrical circuits with different component parameters. The method was tested for two integration methods implemented in the simulation program SPICE (Trapez and Gear). For each type of circuit, a custom network was trained. Experimental simulations showed that for well-defined problems with a sufficiently trained network, the method allows in most cases reducing the total number of iteration steps performed by the algorithm during the simulation computation. Applications of this method, drawbacks, and possible further optimizations are also discussed.

On exponential fitting of finite difference methods for heat equations

In this article, a new kind of finite difference scheme that is exponentially fitted, inspired from Fourier analysis, for a fourth space derivative was developed for solving diffusion problems. Dispersion relation and local truncation error of the method were discussed. Stability analysis of the method revealed that it is conditionally stable. Compared to the corresponding fourth order classical scheme in the literature, the proposed scheme is efficient and accurate. Mathematics Subject Classification (2020): 65M06, 65N06. Keywords: Exponential fitting, Finite difference, Local truncation error, Heat equations.

Solving the Inter-Ring Distances Optimization Problem for Pentapolar and Sextopolar Concentric Ring Electrodes Based on the Negligible Dimensions Model of the Electrode

Concentric ring electrodes are noninvasive and wearable sensors for electrophysiological measurement capable of estimating the surface Laplacian (second spatial derivative of surface potential) at each electrode. Previously, progress was made toward optimization of inter-ring distances (distances between the recording surfaces of a concentric ring electrode), maximizing the accuracy of the surface Laplacian estimate based on the negligible dimensions model of the electrode. However, this progress was limited to tripolar (number of concentric rings n equal to 2) and quadripolar (n = 3) electrode configurations only. In this study, the inter-ring distances optimization problem is solved for pentapolar (n = 4) and sextopolar (n = 5) concentric ring electrode configurations using a wide range of truncation error percentiles ranging from 1st to 25th. Obtained results also suggest consistency between all the considered concentric ring electrode configurations corresponding to n ranging from 2 to 5 that may allow estimation of optimal ranges of inter-ring distances for electrode configurations with n ≥ 6. Therefore, this study may inform future concentric ring electrode design for n ≥ 4 which is important since the accuracy of surface Laplacian estimation has been shown to increase with an increase in n.