A Space-Time Interior Penalty Discontinuous Galerkin Method for the Wave Equation

A new higher-order accurate space-time discontinuous Galerkin (DG) method using the interior penalty flux and discontinuous basis functions, both in space and in time, is presented and fully analyzed for the second-order scalar wave equation. Special attention is given to the definition of the numerical fluxes since they are crucial for the stability and accuracy of the space-time DG method. The theoretical analysis shows that the DG discretization is stable and converges in a DG-norm on general unstructured and locally refined meshes, including local refinement in time. The space-time interior penalty DG discretization does not have a CFL-type restriction for stability. Optimal order of accuracy is obtained in the DG-norm if the mesh size h and the time step $$\Delta t$$ Δ t satisfy $$h\cong C\Delta t$$ h ≅ C Δ t , with C a positive constant. The optimal order of accuracy of the space-time DG discretization in the DG-norm is confirmed by calculations on several model problems. These calculations also show that for pth-order tensor product basis functions the convergence rate in the $$L^\infty$$ L ∞ and $$L^2$$ L 2 -norms is order $$p+1$$ p + 1 for polynomial orders $$p=1$$ p = 1 and $$p=3$$ p = 3 and order p for polynomial order $$p=2$$ p = 2 .

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.

Investigation of Finite-Difference Schemes for the Numerical Solution of a Fractional Nonlinear Equation

The article discusses different schemes for the numerical solution of the fractional Riccati equation with variable coefficients and variable memory, where the fractional derivative is understood in the sense of Gerasimov-Caputo. For a nonlinear fractional equation, in the general case, theorems of approximation, stability, and convergence of a nonlocal implicit finite difference scheme (IFDS) are proved. For IFDS, it is shown that the scheme converges with the order corresponding to the estimate for approximating the Gerasimov-Caputo fractional operator. The IFDS scheme is solved by the modified Newton’s method (MNM), for which it is shown that the method is locally stable and converges with the first order of accuracy. In the case of the fractional Riccati equation, approximation, stability, and convergence theorems are proved for a nonlocal explicit finite difference scheme (EFDS). It is shown that EFDS conditionally converges with the first order of accuracy. On specific test examples, the computational accuracy of numerical methods was estimated according to Runge’s rule and compared with the exact solution. It is shown that the order of computational accuracy of numerical methods tends to the theoretical order of accuracy with increasing nodes of the computational grid.

Optimal with respect to accuracy methods for evaluating hypersingular integrals

In this paper we constructed optimal with respect to order quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M), Ω¯ur,γ(Ω,M), Ω=[−1,1]l, l=1,2,…,M=Const, and γ is a real positive number. The functions that belong to classes Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M) have bounded derivatives up to the rth order in domain Ω and derivatives up to the sth order (s=r+⌈γ⌉) in domain Ω∖Γ, Γ=∂Ω. Moduli of derivatives of the vth order (r<v≤s) are power functions of d(x,Γ)−1(1+|lnd(x,Γ)|), where d(x,Γ) is a distance between point x and Γ. The interest in these classes of functions is due to the fact that solutions of singular and hypersingular integral equations are their members. Moreover various physical fields, in particular gravitational and electromagnetic fields belong to these classes as well. We give definitions of optimal with respect to accuracy methods for solving hypersingular integrals. We constructed optimal with respect to order of accuracy quadrature formulas for evaluating one- and multidimensional hypersingular integrals on classes of functions Ωur,γ(Ω,M) and Ω¯ur,γ(Ω,M).

Explicit Stable Finite Difference Methods for Diffusion-Reaction Type Equations

By the iteration of the theta-formula and treating the neighbors explicitly such as the unconditionally positive finite difference (UPFD) methods, we construct a new 2-stage explicit algorithm to solve partial differential equations containing a diffusion term and two reaction terms. One of the reaction terms is linear, which may describe heat convection, the other one is proportional to the fourth power of the variable, which can represent radiation. We analytically prove, for the linear case, that the order of accuracy of the method is two, and that it is unconditionally stable. We verify the method by reproducing an analytical solution with high accuracy. Then large systems with random parameters and discontinuous initial conditions are used to demonstrate that the new method is competitive against several other solvers, even if the nonlinear term is extremely large. Finally, we show that the new method can be adapted to the advection–diffusion-reaction term as well.

On the efficiency of high-order difference schemes for the Schro¨dinger equation

Исследуется ряд двух- и трехслойных разностных схем, построенных на расширенных шаблонах, до восьмого порядка точности для уравнения Шрёдингера. Наряду с многоточечными схемами рассматривается метод коррекции Ричардсона в приложении к схеме четвертого порядка аппроксимации, повышающий порядок точности путем построения линейных комбинаций приближенных решений, полученных на различных вложенных сетках. Проведено сравнение методов по устойчивости, сложности реализации алгоритмов и объему вычислений, необходимых для достижения заданной точности. На основе теоретического анализа и численных экспериментов выявлены методы, наиболее эффективные для практического применения The efficiency of difference methods for solving problems of nonlinear wave optics is largely determined by the order of accuracy. Schemes up to the fourth order of accuracy have the traditional architecture of three-point stencils and standard conditions for the application of algorithms. However, a further increase in the order in the general case is associated with the need to expand the stencils using multipoint difference approximations of the derivatives. The use of such schemes forces formulating additional boundary conditions, which are not present in the differential problem, and leads to the need to invert the matrices of the strip structure, which are different from the traditional tridiagonal ones. An exception is the Richardson correction method, which is aimed at increasing the order of accuracy by constructing special linear combinations of approximate solutions obtained on various nested grids according to traditional structure schemes. This method does not require the formulation of additional boundary conditions and inversion of strip matrices. In this paper, we consider several explicit and implicit multipoint difference schemes up to the eighth order of accuracy for the Schr¨odinger equation. In addition, a simple and double Richardson correction method is also investigated in relation to the classical fourth-order scheme. A simple correction raises the order to sixth and a double correction to eighth. This large collection of schemes is theoretically compared in terms of their properties such as the order of approximation, stability, the complexity of the implementation of a numerical algorithm, and the amount of arithmetic operations required to achieve a given accuracy. The theoretical analysis is supplemented by numerical experiments on the selected test problem. The main conclusion drawn from the research results is that of all the considered schemes, the Richardson-corrected scheme is the most preferable in terms of the investigated properties

Age-structured delayed SIPCV epidemic model of HPV and cervical cancer cells dynamics I. Numerical method

The numerical method for simulation dynamics of nonlinear epidemic model of age-structured sub-populations of susceptible, infectious, precancerous and cancer cells and unstructured population of human papilloma virus (HPV) is developed (SIPCV model). Cell population dynamics is described by the initial-boundary value problem for the delayed semi-linear hyperbolic equations with age- and time-dependent coefficients and HPV dynamics is described by the initial problem for nonlinear delayed ODE. The model considers two time-delay parameters: the time between viral entry into a target susceptible cell and the production of new virus particles, and duration of the first stage of delayed immune response to HPV population growing. Using the method of characteristics and method of steps we obtain the exact solution of the SIPCV epidemic model in the form of explicit recurrent formulae. The numerical method designed for this solution and used the trapezoidal rule for integrals in recurrent formulae has a second order of accuracy. Numerical experiments with vanished mesh spacing illustrate the second order of accuracy of numerical solution with respect to the benchmark solution and show the dynamical regimes of cell-HPV population with the different phase portraits.

Numerical Modeling of Self-Consistent Dynamics of Shallow and Ground Waters

A mathematical and numerical model of the joint dynamics of shallow and ground waters has been built, which takes into account the nonlinear dynamics of a liquid, water absorption from the surface into the ground, filtration currents in the ground, and water seepage from the ground back to the surface. The dynamics of shallow waters is described by the Saint-Venant equations, taking into account the spatially inhomogeneous distributions of the terrain, the coefficients of bottom friction and infiltration, as well as non-stationary sources and flows of water. For the numerical integration of Saint-Venant’s equations, the well-tested CSPH-TVD method of the second order of accuracy is used, the parallel CUDA algorithm of which is implemented as a software package “EcoGIS-Simulation” for high-performance computing on supercomputers with graphic coprocessors (GPU). The dynamics of groundwater is described by the nonlinear Bussensk equation, generalized to the case of a spatially inhomogeneous distribution of the parameters of the porous medium and the surface of the aquiclude (the boundary between water-permeable and low-permeable soils). The numerical solution of this equation is built on the basis of a finite-difference scheme of the second order of accuracy, the CUDA algorithm of which is integrated into the calculation module of the “EcoGIS-Simulation” software package and is consistent with the main stages of the CSPH-TVD method. The relative deviation of the numerical solution from the exact solution of the nonlinear Boussinesq equation does not exceed 10−4–10−5. The paper compares the results of numerical modeling of the dynamics of groundwater with analytical solutions of the linearized Bussensk equation used as calculation formulas in the methods for predicting the level of groundwater in the vicinity of water bodies. It is shown that the error of these methods is several percent even for the simplest case of a plane-parallel flow of groundwater with a constant backwater. Based on the results obtained, it was concluded that the proposed method for numerical modeling of the joint dynamics of surface and ground waters can be more versatile and efficient (it has significantly better accuracy and productivity) in comparison with the existing methods for calculating flooding zones, especially for hydrodynamic flows with complex geometry and nonlinear interaction of counter fluid flows arising during seasonal floods during flooding of vast land areas.

Reconstructed Interpolating Differential Operator Method with Arbitrary Order of Accuracy for the Hyperbolic Equation

We propose a family of multi-moment methods with arbitrary orders of accuracy for the hyperbolic equation via the reconstructed interpolating differential operator (RDO) approach. Reconstruction up to arbitrary order can be achieved on a single cell from properly allocated model variables including spatial derivatives of varying orders. Then we calculate the temporal derivatives of coefficients of the reconstructed polynomial and transform them into the temporal derivatives of the model variables. Unlike the conventional multi-moment methods which evolve different types of moments by deriving different equations, RDO can update all derivatives uniformly via a simple linear transform more efficiently. Based on difference in introducing interaction from adjacent cells, the central RDO and the upwind RDO are proposed. Both schemes enjoy high-order accuracy which is verified by Fourier analysis and numerical experiments.

Analysis of chosen numerical methods for the application in high order interior ballistics simulations

Considering constant development of the interior ballistics, along with new gun and ammunition designs, the necessity of in-depth analysis of the shot event is continuously increasing. Numerical simulations of interior ballistics problems are useful for optimising new designs or explaining complex issues, regarding performance instabilities and catastrophic failures. With the rise of the computing power, there is a significant urge to drive the numerical errors towards machine zero. This goal demands using methods of high order of accuracy in both space and time. Current methods allow to achieve an arbitrary order of numerical accuracy, thus allowing to shift the focus towards sophistication of the mathematical model of the studied phenomenon. Therefore, in this work, some numerical schemes, in context of finite volume method, are reviewed and studied using well established test problems. The results of the presented analysis are meant to become the basis for future development of a high order numerical scheme for simulation of interior ballistics problems.