| 1. |
- Persson, Ingemar, et al.
(författare)
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On the convergence of multigrid methods for flow problems
- 1999
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Ingår i: Electronic Transactions on Numerical Analysis. - 1068-9613. ; 8, s. 46-87
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Tidskriftsartikel (refereegranskat)abstract
- We prove two theorems on the residual damping in multigrid methods when solving convection dominated diffusion equations and shock wave problems, discretized by the streamline diffusion finite element method. The first theorem shows that a V-cycle, including sufficiently many pre and post smoothing steps, damps the residual in LIloc for a constant coefficient convection problem with small diffusion in two space dimensions, without the assumption that the coarse grid is sufficiently fine. The proof is based on discrete Green's functions for the smoothing and correction operators on a uniform unbounded mesh aligned with the characteristic. The second theorem proves a similar result for a certain continuous version of a two grid method, with Isotropic artificial diffusion, applied to a two dimensional Burgers shock wave problem. We also present numerical experiments that verify the residual damping dependence on the equation, the choice of artificial diffusion and the number of smoothing steps. In particular numerical experiments show improved convergence of the multigrid method, with damped Jacobi smoothing steps, for the compressible Navier-Stokes equations in two space dimensions by using the theoretically suggested exponential increase of the number of smoothing steps on coarser meshes, as compared to the same amount of work with constant number of smoothing steps on each level.
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| 2. |
- Goodman, Jonathan, et al.
(författare)
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A remark on the stability of viscous shock-waves
- 1994
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Ingår i: SIAM Journal on Mathematical Analysis. - PHILADELPHIA : SIAM PUBLICATIONS. - 0036-1410 .- 1095-7154. ; 25:6, s. 1463-1467
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Tidskriftsartikel (refereegranskat)abstract
- Recently, Szepessy and Xin gave a new proof of stability of viscous shock waves. A curious aspect of their argument is a possible disturbance of zero mass, but log(t)t-1/2 amplitude in the vicinity of the shock wave. This would represent a previously unobserved phenomenon. However, only an upper bound is established in their proof. Here, we present an example of a system for which this phenomenon can be verified by explicit calculation. The disturbance near the shock is shown to be precisely of order t-1/2 in amplitude.
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| 3. |
- Hansbo, Peter F G, 1959, et al.
(författare)
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A velocity-pressure streamline diffusion finite element method for the incompressible Navier-Stokes equations
- 1990
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Ingår i: Computer Methods in Applied Mechanics and Engineering. - LAUSANNE : Elsevier BV. ; 84:2, s. 175-192
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Tidskriftsartikel (refereegranskat)abstract
- In this paper a streamline diffusion finite element method is introduced for the time-dependent incompressible Navier-Stokes equations in a bounded domnain in R^2 and R^3 in the case of high Reynolds number flow. An error estimate is proved and numerical results are given. The method is based on a mixed velocity-pressure formulation using the same finite element discretization of space-time for the velocity and the pressure spaces, which consists of piecewise linear functions, together with certain least-squares modifications of the Galerkin variational formulation giving added stability without sacrificing accuracy.
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| 4. |
- Hoel, Hakon, et al.
(författare)
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Classical langevin dynamics derived from quantum mechanics
- 2020
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Ingår i: Discrete and continuous dynamical systems. Series B. - : AMER INST MATHEMATICAL SCIENCES-AIMS. - 1531-3492 .- 1553-524X. ; 25:10, s. 4001-4038
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Tidskriftsartikel (refereegranskat)abstract
- The classical work by Zwanzig [J. Stat. Phys. 9 (1973) 215-220] derived Langevin dynamics from a Hamiltonian system of a heavy particle coupled to a heat bath. This work extends Zwanzig's model to a quantum system and formulates a more general coupling between a particle system and a heat bath. The main result proves, for a particular heat bath model, that ab initio Langevin molecular dynamics, with a certain rank one friction matrix determined by the coupling, approximates for any temperature canonical quantum observables, based on the system coordinates, more accurately than any Hamiltonian system in these coordinates, for large mass ratio between the system and the heat bath nuclei.
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| 5. |
- Huang, Xin, et al.
(författare)
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Canonical mean-field molecular dynamics derived from quantum mechanics
- 2022
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Ingår i: ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS. - : EDP Sciences. - 2822-7840 .- 2804-7214. ; 56:6, s. 2197-2238
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Tidskriftsartikel (refereegranskat)abstract
- Canonical quantum correlation observables can be approximated by classical molecular dynamics. In the case of low temperature the ab initio molecular dynamics potential energy is based on the ground state electron eigenvalue problem and the accuracy has been proven to be O(M-1), provided the first electron eigenvalue gap is sufficiently large compared to the given temperature and M is the ratio of nuclei and electron masses. For higher temperature eigenvalues corresponding to excited electron states are required to obtain O(M-1) accuracy and the derivations assume that all electron eigenvalues are separated, which for instance excludes conical intersections. This work studies a mean-field molecular dynamics approximation where the mean-field Hamiltonian for the nuclei is the partial trace h := Tr(He-beta H)/Tr(e(-beta H)) with respect to the electron degrees of freedom and H is the Weyl symbol corresponding to a quantum many body Hamiltonian (sic). It is proved that the mean-field molecular dynamics approximates canonical quantum correlation observables with accuracy O(M-1 + t epsilon(2)), for correlation time t where epsilon(2) is related to the variance of mean value approximation h. Furthermore, the proof derives a precise asymptotic representation of the Weyl symbol of the Gibbs density operator using a path integral formulation. Numerical experiments on a model problem with one nuclei and two electron states show that the mean-field dynamics has similar or better accuracy than standard molecular dynamics based on the ground state electron eigenvalue.
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| 6. |
- Johnson, Claes, et al.
(författare)
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Adaptive finite element methods for conservation laws based on a posteriori error estimates
- 1995
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Ingår i: Communications on Pure and Applied Mathematics. - NEW YORK : John Wiley & Sons. - 0010-3640 .- 1097-0312. ; 48:3, s. 199-234
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Tidskriftsartikel (refereegranskat)abstract
- We prove a posteriori error estimates for a finite element method for systems of strictly hyperbolic conservation laws in one space dimension, and design corresponding adaptive methods. The proof of the a posteriori error estimates is based on a strong stability estimate for an associated dual problem, together with the Galerkin orthogonality of the finite-element method. The strong stability estimate uses the entropy condition for the system in an essential way.
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| 7. |
- Johnson, Claes, et al.
(författare)
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On the convergence of shock-capturing streamline diffusion finite element methods for hyperbolic conservation laws
- 1990
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Ingår i: Mathematics of Computation. - PROVIDENCE : American Mathematical Society (AMS). - 0025-5718 .- 1088-6842. ; 54:189, s. 107-129
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Tidskriftsartikel (refereegranskat)abstract
- We extend our previous analysis of streamline diffusion finite element methods for hyperbolic systems of conservation laws to include a shock-capturing term adding artificial viscosity depending on the local absolute value of the residual of the finite element solution and the meh size. With this term present, we prove a maximum norm bound for finite element solutionsof Burgers' equation an thus complete an earlier convergence proof for this equation. We further prove, using entropy variables, that a strong limit of finite element solutions is a weak solution of the system of conservation laws and satisfies the entropy inequality asociated with the entropy variables. Results of some numerical experiments for the time-dependent compressible Euler equations in two dimensions are also reported.
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| 8. |
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| 9. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Adaptive random fourier features with metropolis sampling
- 2019
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Ingår i: Foundations of Data Science. - : American Institute of Mathematical Sciences. - 2639-8001. ; 0:0, s. 0-0
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Tidskriftsartikel (refereegranskat)abstract
- The supervised learning problem todetermine a neural network approximation $\mathbb{R}^d\ni x\mapsto\sum_{k=1}^K\hat\beta_k e^{{\mathrm{i}}\omega_k\cdot x}$with one hidden layer is studied asa random Fourier features algorithm. The Fourier features, i.e., the frequencies $\omega_k\in\mathbb{R}^d$,are sampled using an adaptive Metropolis sampler.The Metropolis test accepts proposal frequencies $\omega_k'$, having corresponding amplitudes $\hat\beta_k'$, with the probability$\min\big\{1, (|\hat\beta_k'|/|\hat\beta_k|)^\gamma\big\}$,for a certain positive parameter $\gamma$, determined by minimizing the approximation error for given computational work.This adaptive, non-parametric stochastic method leads asymptotically, as $K\to\infty$, to equidistributed amplitudes $|\hat\beta_k|$, analogous to deterministic adaptive algorithms for differential equations. The equidistributed amplitudes are shown to asymptotically correspond to the optimal density for independent samples in random Fourier features methods.Numerical evidence is provided in order to demonstrate the approximation properties and efficiency of the proposed algorithm. The algorithm is testedboth on synthetic data and a real-world high-dimensional benchmark.
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| 10. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Canonical quantum observables for molecular systems approximated by ab initio molecular dynamics
- 2018
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Ingår i: Annales Henri Poincaré. - : Springer Nature. - 1424-0637 .- 1424-0661. ; 19, s. 2727-2781
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Tidskriftsartikel (refereegranskat)abstract
- It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.
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| 11. |
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| 12. |
- Kammonen, Aku, 1984-, et al.
(författare)
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Canonical quantum observables for molecular systems approximated by ab inition molecular dynamics
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- It is known that ab initio molecular dynamics based on the electron ground state eigenvaluecan be used to approximate quantum observables in the canonical ensemble when the temperature is low compared tothe first electron eigenvalue gap. This work proves that a certain weighted average of the different ab initio dynamics, corresponding to each electron eigenvalue, approximates quantum observables for any temperature.The proof uses the semi-classical Weyl law to show thatcanonical quantum observables of nuclei-electron systems, based on matrix valued Hamiltonian symbols, can be approximated by ab initio molecular dynamics with the error proportional to the electron-nuclei mass ratio. The resultincludes observables that depend on correlations in time. A combination of the Hilbert-Schmidt inner product for quantum operators and Weyl's lawshows that the error estimate holds %for observables and Hamiltonian symbols that have three and five bounded derivatives, respectively, provided the electron eigenvalues are distinct for any nuclei positionand the observables are in diagonal form with respect to the electron eigenstates.
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| 13. |
- Kammonen, Aku, 1984-, et al.
(författare)
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COMPUTATIONAL ALGORITHMS FOR CANONICAL ENSEMBLE OBSERVABLES
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We consider canonical ensembles of molecular systems. We propose four numerical algorithms for efficient computation of the canonical ensemble molecular dynamics observables. The four algorithms can each be efficient in different situations. For example in low temperatures we can make use of the fact that the lowest electron energy levels contributes most to the observable.
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| 14. |
- Kammonen, Aku, 1984-
(författare)
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Numerical algorithms for high dimensional integration with application to machine learning and molecular dynamics
- 2021
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Doktorsavhandling (övrigt vetenskapligt/konstnärligt)abstract
- This thesis contains results on high dimensional integration with two papers, paper I and paper II, presenting applications in machine learning and two papers, paper III and paper IV, presenting applications to molecular dynamics.In paper I we present algorithms based on a Metropolis test for training shallow neural networks with trigonometric activation functions. Numerical experiments are performed on both synthetic and real data. The trigonometric activation function gives access to the Fourier transform and its inverse transform. The algorithms gives equidistributed amplitudes.In paper II we derive smaller generalization error for deep residual neural networks compared to shallow ones. An algorithm that builds the residual neural network layer by layer based on an algorithm from paper I is presented both as a stand alone algorithm as well as a pre-step for a global optimizer like Stochastic gradient descent or Adam. Numerical test are performed with promising results.In paper III we make use of the semiclassical Weyl law to show that canonical quantum observables can be approximated by molecular dynamics with an error rate proportional to the electron-nuclei mass ratio. Numerical experiments are presented that confirms the expected theoretical result.In paper IV we consider canonical ensembles of molecular systems. We propose four numerical algorithms for efficient computation of the canonical ensemble molecular dynamics observables. The four algorithms can each be efficient in different situations. For example in low temperatures we can make use of the fact that the lowest electron energy levels contributes most to the observable. The work is an extension of the results in paper III.
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| 15. |
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| 16. |
- Kammonen, Aku, 1984-, et al.
(författare)
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SMALLER GENERALIZATION ERROR DERIVED FOR DEEP COMPARED TO SHALLOW RESIDUAL NEURAL NETWORKS
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- Estimates of the generalization error are proved for a residual neural network with $L$ random Fourier features layers $\bar z_{\ell+1}=\bar z_\ell + \mathrm{Re}\sum_{k=1}^K\bar b_{\ell k}e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}+\mathrm{Re}\sum_{k=1}^K\bar c_{\ell k}e^{\mathrm{i}\omega'_{\ell k}\cdot x}$. An optimal distribution for the frequencies $(\omega_{\ell k},\omega'_{\ell k})$ of the random Fourier features $e^{\mathrm{i}\omega_{\ell k}\bar z_\ell}$ and $e^{\mathrm{i}\omega'_{\ell k}\cdot x}$ is derived. This derivation is based on the corresponding generalization error for the approximation of the function values $f(x)$. The generalization error turns out to be smaller than the estimate ${\|\hat f\|^2_{L^1(\mathbb{R}^d)}}/{(LK)}$ of the generalization error for random Fourier features with one hidden layer and the same total number of nodes $LK$, in the case the $L^\infty$-norm of $f$ is much less than the $L^1$-norm of its Fourier transform $\hat f$. This understanding of an optimal distribution for random features is used to construct a new training method for a deep residual network that shows promising results.
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| 17. |
- Lindholm, Love, 1974-, et al.
(författare)
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A mean field game model of an electricity market with consumers minimizing energy cost through dynamic battery usage
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- This work contains a model of an electricity market consisting of consumers who own batteries that they charge and discharge in an optimal way. The goal of each individual customer is to minimize their total electricity cost, not by changing how much they consume, but by utilizing an optimal strategy for their battery usage. For each consumer we therefore have a value function. Since all consumers are assumed to be equal, their associated value functions are also equal. The optimization problem to determine the optimal battery usage depends on the electricity price, which in turn depends on the total electricity consumption. The consumption is given as a solution to a Kolmogorov forward equation, which involves the battery usage. Hence the Hamilton-Jacobi-Bellman and Kolmogorov equations need to be solved together as a coupled system of PDEs. We devise a numerical scheme for this system and show some simulations. We also prove a result on the existence and uniqueness of solutions to the system of PDEs.
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| 18. |
- Lindholm, Love, 1974-, et al.
(författare)
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Local volatility calibration with optimal control in a Hamiltonian framework
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Annan publikation (övrigt vetenskapligt/konstnärligt)abstract
- We formulate the calibration of a local volatility function that makes a solution to Dupie's equation match market data as an optimal control problem for which optimality conditions are given by a Hamiltonian system. Regularization is added by mollifying the Hamiltonian functional in this system. We have direct access to the Jacobian matrix of the Hamiltonian system, and can therefore employ a Newton based method in the solving phase, whereas other studies tend to use gradient based methods or quasi Newton algorithms. We illustrate our method by calibrating a volatility function to market data on the Euro Stoxx 50 index and find that our algorithm is both accurate and robust.
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| 19. |
- Plechác, P., et al.
(författare)
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The classical limit of quantum observables in the conservation laws of fluid dynamics
- 2019
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Ingår i: Communications in Mathematical Sciences. - : International Press of Boston, Inc.. - 1539-6746 .- 1945-0796. ; 17:8, s. 2191-2221
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Tidskriftsartikel (refereegranskat)abstract
- In the classical work by Irving and Zwanzig [J.H. Irving and R.W. Zwanzig, J. Chem. Phys., 19, 1173-1180, 1951] it has been shown that quantum observables for macroscopic density, momentum and energy satisfy the conservation laws of fluid dynamics. In this work we derive the corresponding classical molecular dynamics limit by extending Irving and Zwanzig's result to matrix-valued potentials for a general quantum particle system. The matrix formulation provides the classical limit of the quantum observables in the conservation laws also in the case where the temperature is large compared to the electron eigenvalue gaps. The classical limit of the quantum observables in the conservation laws is useful in order to determine the constitutive relations for the stress tensor and the heat flux by molecular dynamics simulations. The main new steps to obtain the molecular dynamics limit are: (i) to approximate the dynamics of quantum observables accurately by classical dynamics, by diagonalizing the Hamiltonian using a nonlinear eigenvalue problem, (ii) to define the local energy density by partitioning a general potential, applying perturbation analysis of the electron eigenvalue problem, (iii) to determine the molecular dynamics stress tensor and heat flux in the case of several excited electron states, and (iv) to construct the initial particle phase-space density as a local grand canonical quantum ensemble determined by the initial conservation variables.
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| 20. |
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| 21. |
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| 22. |
- Szepessy, Anders, 1960-
(författare)
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Convergence of a streamline diffusion finite element method for scalar conservation laws with boundary conditions
- 1991
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Ingår i: Mathematical Modelling and Numerical Analysis. - MONTROUGE CEDEX : Dunod Editeur. - 0764-583X .- 1290-3841. ; 25:6, s. 749-783
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Tidskriftsartikel (refereegranskat)abstract
- A higher order accurate shock-capturing streamline diffusion finite element method for general scalar conservation laws is analysed; convergence towards the unique solution is proved for several space dimensions with initial and boundary conditions, using a uniqueness theorem for measure valued solutions. Furthermore, some numerical results are given.
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| 23. |
- Szepessy, Anders, 1960-
(författare)
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Dynamics and stability of a weak detonation wave
- 1999
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Ingår i: Communications in Mathematical Physics. - NEW YORK : Springer-Verlag New York. - 0010-3616 .- 1432-0916. ; 202:3, s. 547-569
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Tidskriftsartikel (refereegranskat)abstract
- One dimensional weak detonation waves of a basic reactive shock wave model are proved to be nonlinearly stable, i.e. initially perturbed waves tend asymptotically to translated weak detonation waves. This model system was derived as the low Math number limit of the one component reactive Navier-Stokes equations by Majda and Roytburd [SIAM J. Sci. Stat. Comput. 43, 1086-1118 (1983)], and its weak detonation waves have been numerically observed as stable. The analysis shows in particular the key role of the new nonlinear dynamics of the position of the shock wave, The shock translation solves a nonlinear integral equation, obtained by Green's function techniques, and its solution is estimated by observing that the kernel can be split into a dominating convolution operator and a remainder. The inverse operator of the convolution and detailed properties of the traveling wave reduce, by monotonicity, the remainder to a small L-1 perturbation.
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| 24. |
- Szepessy, Anders, 1960-
(författare)
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Langevin molecular dynamics derived from Ehrenfest dynamics
- 2011
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Ingår i: Mathematical Models and Methods in Applied Sciences. - : World Scientific. - 0218-2025. ; 21:11, s. 2289-2334
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Tidskriftsartikel (refereegranskat)abstract
- Stochastic Langevin molecular dynamics for nuclei is derived from the Ehrenfest Hamiltonian system (also called quantum classical molecular dynamics) in a KacZwanzig setting, with the initial data for the electrons stochastically perturbed from the ground state and the ratio M of nuclei and electron mass tending to infinity. The Ehrenfest nuclei dynamics is approximated by the Langevin dynamics with accuracy o(M-1/2) on bounded time intervals and by o(1) on unbounded time intervals, which makes the small O(M -1/2) friction and o(M-1/2) diffusion terms visible. The initial electron probability distribution is a Gibbs density at low temperature, motivated by a stability and consistency argument. The diffusion and friction coefficients in the Langevin equation satisfy the Einstein's fluctuationdissipation relation.
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| 25. |
- Szepessy, Anders, 1960-
(författare)
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Measure valued solutions of scalar conservation laws with boundary conditions
- 1989
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Ingår i: Archive for Rational Mechanics and Analysis. - NEW YORK : Springer-Verlag New York. - 0003-9527 .- 1432-0673. ; 107:2, s. 181-193
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Tidskriftsartikel (refereegranskat)abstract
- We define a solution concept for measure-valued solutions to scalar conservation laws with initial conditions and boundary conditions and prove a uniqueness theorem for such solutions. This result may be used to prove convergence, towards the unique solution, for approximate solutions which are uniformly bounded in L∞, weakly consistent with certain entropy inequalities and strongly consistent with the initial condition, i.e. without using derivative estimates. As an example convergence of a finite element method is demonstrated.
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| 26. |
- Szepessy, Anders, 1960-, et al.
(författare)
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Nonlinear stability of viscous shock waves
- 1993
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Ingår i: Archive for Rational Mechanics and Analysis. - NEW YORK : Springer-Verlag New York. - 0003-9527 .- 1432-0673. ; 122:1, s. 53-103
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Tidskriftsartikel (refereegranskat)
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| 27. |
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| 28. |
- Szepessy, Anders, 1960-
(författare)
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On the stability of finite element methods for shock waves
- 1992
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Ingår i: Communications on Pure and Applied Mathematics. - NEW YORK : John Wiley & Sons. - 0010-3640 .- 1097-0312. ; 45:8, s. 923-946
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Tidskriftsartikel (refereegranskat)abstract
- this paper we study the large time asymptotic stability of solutions for systems of nonlinear viscous conservation laws of the form (1:1) u t + f(u) x = u xx ; x 2 R I ; t ? 0 ; u 2 R I u(\Delta; 0) = u 0 (\Delta) : We treat systems which are strictly hyperbolic. Such systems possess a smooth travelling wave solution, which is called a viscous p-shock wave solution, u(x; t) = OE(x \Gamma oet) x!\Sigma1 OE(x) = u \Sigma ; provided that the shock strength ffl j ju + \Gamma u \Gamma j is small [19], the constant states u \Sigma and the wave speed oe are related by the Rankine-Hugoniot condition (1:3a) f(u \Gamma ) \Gamma f(u+ ) = oe(u \Gamma \Gamma u+ )
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| 29. |
- Szepessy, Anders, 1960-, et al.
(författare)
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Stability of rarefaction waves in viscous media
- 1996
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Ingår i: Archive for Rational Mechanics and Analysis. - NEW YORK : Springer-Verlag New York. - 0003-9527 .- 1432-0673. ; 133:3, s. 249-298
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Tidskriftsartikel (refereegranskat)abstract
- We study the time-asymptotic behavior of weak rarefaction waves of systems of conservation laws describing one-dimensional viscous media, with strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, ''Burgers'' rarefaction wave, for initial perturbations w(o) with small mass and localized as w(o)(x)= O(\x\(-1)). The proof proceeds by iteration of a pointwise ansatz for the error, using integral representations of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our method, we also obtain rates of decay and detailed pointwise estimates for the error. This pointwise method has been used successfully in studying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized equations and error waves of unbounded mass O(log(t)). These ''diffusion waves'' have amplitude O(t(-1/2) log t) in linear degenerate transversal fields and O(t(-1/2)) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof
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