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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., Comment: 98 pages, 44 figures, 21 tables. arXiv admin note: text overlap with arXiv:2409.09933

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 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 multiplication. At the moment, the main interface of software libraries for high-performance computing is BLAS. This paper presents an effective method for integration the standard functions of the BLAS interface into the implementation of these numerical methods. The calculated matrices are small matrices; at the same time, the proposed implementation makes it possible to effectively use existing JIT technologies. The proposed approach immediately operates on AoS, which makes it possible to efficiently calculate flux, source and non-conservative terms without need to carry out transposition. The obtained computational costs allowed to conclude 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 naive implementations by several times. At the same time, the complexity of developing an implementation based on the approach proposed in this work does not exceed the complexity of developing a naive optimized implementation., Comment: 33 pages, 5 figures, 13 tables. arXiv admin note: text overlap with arXiv:2409.09911

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., Comment: 44 pages, 12 figures, 7 tables

The space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter was used to simulate non-stationary compressible multicomponent reactive flows. The multicomponent composition of the reacting medium and the reactions occurring in it were described by expanding the original system of Euler equations by a system of non-stationary convection-reaction equations. The use of this method to simulate high stiff problems associated with reactions occurring in a multicomponent medium requires the use of the adaptive change in the time step. The solution of the classical problem related to the formation and propagation of a ZND detonation wave is carried out. It was shown that the space-time adaptive ADER finite element DG method with a posteriori correction technique of solutions on subcells by the finite-volume ADER-WENO limiter can be used to simulate flows without using of splitting in directions and fractional step methods., Comment: 20 pages, 11 figures, 2 tables

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., Comment: 52 pages, 36 figures, 3 tables

The Belle II experiment at the SuperKEKB electron-positron collider aims to collect an unprecedented data set of $50~{\rm ab}^{-1}$ to study $CP$-violation in the $B$-meson system and to search for Physics beyond the Standard Model. SuperKEKB is already the world's highest-luminosity collider. In order to collect the planned data set within approximately one decade, the target is to reach a peak luminosity of $\rm 6 \times 10^{35}~cm^{-2}s^{-1}$ by further increasing the beam currents and reducing the beam size at the interaction point by squeezing the betatron function down to $\beta^{*}_{\rm y}=\rm 0.3~mm$. To ensure detector longevity and maintain good reconstruction performance, beam backgrounds must remain well controlled. We report on current background rates in Belle II and compare these against simulation. We find that a number of recent refinements have significantly improved the background simulation accuracy. Finally, we estimate the safety margins going forward. We predict that backgrounds should remain high but acceptable until a luminosity of at least $\rm 2.8 \times 10^{35}~cm^{-2}s^{-1}$ is reached for $\beta^{*}_{\rm y}=\rm 0.6~mm$. At this point, the most vulnerable Belle II detectors, the Time-of-Propagation (TOP) particle identification system and the Central Drift Chamber (CDC), have predicted background hit rates from single-beam and luminosity backgrounds that add up to approximately half of the maximum acceptable rates., Comment: 28 pages, 17 figures, 9 tables (revised); in v3, we corrected Table 5 units for the measured fast neutron fluence by the TPC detectors from x10^9 n_eq/cm^2 to x10^8 n_eq/cm^2, which was a typo in the previous versions; also, we added the value of the luminosity for Tables 5 and 6