arXiv subjects

I. S. Popov

Publications and source records attributed to I. S. Popov.

8 recordsLinked to original sources

arXiv2026

Theory and internal structure of ADER-DG method for partial differential equations

Highly accurate stability boundary values for the ADER-DG method are obtained for arbitrary degrees $N$ of basis polynomials. In the linear case, stability is violated precisely when one of the matrix eigenvalues reaches $λ= -1$, regardless of the phase $θ$. A rigorous mathematical framework for the stability is developed. The stability condition is significantly simplified, reducing it to the problem of calculating the roots of polynomials in the Courant number $\mathrm{CFL}$. The maximum of the Courant numbers $\mathrm{CFL}_{\rm max}(N)$ are calculated. These results are new and very convenient for practical use. A comparison of the obtained results with existing results reveals differences that may be significant for the selection of calculation parameters, especially for high degrees $N$. It is shown that widely used existing estimates $\mathrm{CFL}_{\rm max}(N) \propto 1/(2N+1)$ are overestimated. An interesting qualitative asymptotic $\mathrm{CFL}_{\rm max}(N) \propto 1/(N+1)^{2}$ is obtained. A rigorous direct proof of the approximation is presented. Approximation orders $p = N+1$ for arbitrary degrees $N$ are rigorously derived. A set of numerical experiments is carried out to apply the ADER-DG method to solving both a linear advection equation and an Euler system of equations. The results obtained in these calculations confirm the theoretical results well. In particular, an excess of the Courant number over the $\mathrm{CFL}_{\rm max}(N)$ by even 1% in the linear case immediately leads to significant instability of the numerical solution. The obtained estimates of the boundary Courant number in the nonlinear case are somewhat underestimated -- by no more than 5%, which is due to the diffusivity and stability of the approximate Riemann solver. Empirical convergence orders are obtained, which are in good agreement with the theoretical results.

physics.flu-dyn↗

arXiv2026

Improving the local solution of the DG predictor of the ADER-DG method for solving systems of ordinary differential equations and its applicability to systems of differential-algebraic equations

Improved local numerical solution for the ADER-DG numerical method with a local DG predictor for solving the initial value problem for a first-order ODE system is proposed. The improved local numerical solution demonstrates convergence orders of one higher than the convergence order of the local numerical solution of the original ADER-DG numerical method and has the property of continuity at grid nodes. Rigorous proofs of the approximation orders of the local numerical solution and the improved local numerical solution are presented. Obtaining the proposed improved local numerical solution does not require significant changes to the structure of the ADER-DG numerical method. Therefore, all conclusions regarding the convergence orders of the numerical solution at grid nodes, the resulting superconvergence, and the high stability of the ADER-DG numerical method remain unchanged. A wide range of applications of the ADER-DG numerical method is presented for solving specific initial value problems for ODE systems for a wide range of polynomial degrees. The obtained results provide strong confirmation for the developed rigorous theory. The improved local numerical solution is shown to exhibit both higher accuracy and improved smoothness and point-wise comparability. Empirical convergence orders of all individual numerical solutions were calculated for a wide range of error norms, which well agree with the expected convergence orders. The rigorous proof, based on the $ε$-embedding method, of the applicability of the ADER-DG numerical method with a local DG predictor to solving DAE systems is presents.

math.NA↗

arXiv2026

Theory and internal structure of ADER-DG method for ordinary differential equations

Investigation of the approximation properties, convergence, and stability of the ADER-DG method for solving an ODE system is carried out. The ADER-DG method is $A$- and $AN$-stable, $L$-stable, $B$- and $BN$-stable, and algebraically stable. Several other relations useful for an application and implementation of the ADER-DG method are proved. Applications of the ADER-DG method demonstrated compliance with the expected theoretical results.

math.NA↗

arXiv2025

The effective use of BLAS interface for implementation of finite-element ADER-DG and finite-volume ADER-WENO methods

Numerical methods of the ADER family, in particular finite-element ADER-DG and finite-volume ADER-WENO methods, are among the most accurate numerical methods for solving quasilinear hyperbolic PDE systems. The internal structure of ADER-DG and ADER-WENO numerical methods contains a large number of basic linear algebra operations related to matrix multiplications. The main interface of software libraries for matrix multiplications for high-performance computing is BLAS. An effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods is presented. The calculated matrices are small matrices; and this allows to use effectively JIT technologies. The proposed approach immediately operates on AoS, which allows to efficiently calculate flux, source and non-conservative terms without transposition. The obtained computational costs demonstrated that the effective implementation, based on the use of the JIT functions of the BLAS, outperformed both the implementation based on the general BLAS functions and the vanilla implementations by several orders of magnitude. The complexity of developing an implementation based on the proposed approach does not exceed the complexity of developing a vanilla implementation. Performance analysis using roofline partly explains the observed features of the decreasing of computational costs.

physics.flu-dyn↗

arXiv2024

High order ADER-DG method with local DG predictor for solutions of differential-algebraic systems of equations

A numerical method ADER-DG with a local DG predictor for solving a DAE system has been developed, which was based on the formulation of ADER-DG methods using a local DG predictor for solving ODE and PDE systems. The basis functions were chosen in the form of Lagrange interpolation polynomials with nodal points at the roots of the Radau polynomials, which differs from the classical formulations of the ADER-DG method, where it is customary to use the roots of Legendre polynomials. It was shown that the use of this basis leads to A-stability and L1-stability in the case of using the DAE solver as ODE solver. The numerical method ADER-DG allows one to obtain a highly accurate numerical solution even on very coarse grids, with a step greater than the main characteristic scale of solution variation. The local discrete time solution can be used as a numerical solution of the DAE system between grid nodes, thereby providing subgrid resolution even in the case of very coarse grids. The classical test examples were solved by developed numerical method ADER-DG. With increasing index of the DAE system, a decrease in the empirical convergence orders p is observed. An unexpected result was obtained in the numerical solution of the stiff DAE system -- the empirical convergence orders of the numerical solution obtained using the developed method turned out to be significantly higher than the values expected for this method in the case of stiff problems. It turns out that the use of Lagrange interpolation polynomials with nodal points at the roots of the Radau polynomials is much better suited for solving stiff problems. Estimates showed that the computational costs of the ADER-DG method are approximately comparable to the computational costs of implicit Runge-Kutta methods used to solve DAE systems. Methods were proposed to reduce the computational costs of the ADER-DG method.

math.NA↗

arXiv2024

Arbitrary high order ADER-DG method with local DG predictor for solutions of initial value problems for systems of first-order ordinary differential equations

An adaptation of the arbitrary high order ADER-DG numerical method with local DG predictor for solving the IVP for a first-order non-linear ODE system is proposed. The proposed numerical method is a completely one-step ODE solver with uniform steps, and is simple in algorithmic and software implementations. It was shown that the proposed version of the ADER-DG numerical method is A-stable and L-stable. The ADER-DG numerical method demonstrates superconvergence with convergence order 2N+1 for the solution at grid nodes, while the local solution obtained using the local DG predictor has convergence order N+1. It was demonstrated that an important applied feature of this implementation of the numerical method is the possibility of using the local solution as a solution with a subgrid resolution, which makes it possible to obtain a detailed solution even on very coarse coordinate grids. The scale of the error of the local solution, when calculating using standard representations of single or double precision floating point numbers, using large values of the degree N, practically does not differ from the error of the solution at the grid nodes. The capabilities of the ADER-DG method for solving stiff ODE systems characterized by extreme stiffness are demonstrated. Estimates of the computational costs of the ADER-DG numerical method are obtained.

math.NA↗

arXiv2024

Space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting for multidimensional detonation waves simulation

The space-time adaptive ADER-DG finite element method with LST-DG predictor and a posteriori sub-cell ADER-WENO finite-volume limiting was used for simulation of multidimensional reacting flows with detonation waves. The presented numerical method does not use any ideas of splitting or fractional time steps methods. The modification of the LST-DG predictor has been developed, based on a local partition of the time step in cells in which strong reactivity of the medium is observed. This approach made it possible to obtain solutions to classical problems of flows with detonation waves and strong stiffness, without significantly decreasing the time step. The results obtained show the very high applicability and efficiency of using the ADER-DG-PN method with a posteriori sub-cell limiting for simulating reactive flows with detonation waves. The numerical solution shows the correct formation and propagation of ZND detonation waves. The structure of detonation waves is resolved by this numerical method with subcell resolution even on coarse spatial meshes. The smooth components of the numerical solution are correctly and very accurately reproduced by the numerical method. Non-physical artifacts of the numerical solution, typical for problems with detonation waves, such as the propagation of non-physical shock waves and weak detonation fronts ahead of the main detonation front, did not arise in the results obtained. The results of simulating rather complex problems associated with the propagation of detonation waves in significantly inhomogeneous domains are presented, which show that all the main features of detonation flows are correctly reproduced by this numerical method. It can be concluded that the space-time adaptive ADER-DG-PN method with a posteriori sub-cell ADER-WENO finite-volume limiting is perfectly applicable to simulating complex reacting flows with detonation waves.

physics.flu-dyn↗

arXiv2017

Universal Berezinskii-Kosterlitz-Thouless dynamic scaling in the intermediate timerange in frustrated Heisenberg antiferromagnets on triangular lattice

We investigate non-equilibrium properties of the frustrated Heisenberg antiferromagnets on the triangular lattice. Nonequilibrium critical relaxation of frustrated Heisenberg antiferromagnets shows a dynamic transition (or, at least, sharp crossover) at the same temperature $T_{\rm u}=0.282 J$ as for static properties due to unbinding of $\mathbb{Z}_2$-vortices. We show that starting from the high-temperature initial state, due to presence of $\mathbb{Z}_2$-vortices in the considering system, in a broad temperature range $T<T_{\rm u}$ the dynamic properties in the intermediate time range are similar to those of two-dimensional XY model below Berezinskii-Kosterlitz-Thouless transition. The interaction of $\mathbb{Z}_2$-vortices with spin-wave degrees of freedom does not emerge until rather long times.

cond-mat.str-el↗