284 research outputs found
Asymptotic Analysis and Numerical Approximation of some Partial Differential Equations on Networks
In this thesis, we consider three different model problems on one-dimensional networks with applications in gas, water supply, and district heating networks, as well as bacterial chemotaxis. On each edge of the graph representing the network, the dynamics are described by partial differential equations. Additional coupling conditions at network junctions are needed to ensure basic physical principles and to obtain well-posed systems. Each of the model problems under consideration contains an asymptotic parameter epsilon>0, which is assumed to be small, describing either a singular perturbation, different modeling scales, or different physical regimes.
A central objective of this work is the investigation of the asymptotic behavior of solutions for epsilon going to zero. Moreover, we focus on suitable numerical approximations based on Galerkin methods that are still viable in the asymptotic limit epsilon=0 and preserve the structure and basic properties of the underlying problems.
In the first part, we consider singularly perturbed convection-diffusion equations on networks as well as the corresponding pure transport equations arising in the vanishing diffusion limit for epsilon going to zero, in which the coupling conditions change in number and type. This gives rise to interior boundary layers at network junctions. On a single interval, corresponding asymptotic estimates are well-established. A main contribution is the transfer of these results to networks.
For an appropriate numerical approximation, we propose a hybrid discontinuous Galerkin method which is particularly suitable for dominating convection and coupling at network junctions. An approximation strategy is developed based on layer-adapted meshes, leading to epsilon-uniform error estimates.
The second part is dedicated to a kinetic model of chemotaxis on networks describing the movement of bacteria being influenced by the presence of a chemical substance. Via a suitable scaling the classical Keller-Segel equations can be derived in the diffusion limit. We propose a proper set of coupling conditions that ensure the conservation of mass and lead to a well-posed problem. The local existence of solutions uniformly in the scaling can be established via fixed point arguments. Appropriate a-priori estimates then enable us to rigorously show the convergence of solutions to the diffusion limit. Via asymptotic expansions, we also establish a quantitative asymptotic estimate.
In the last part, we focus on models for gas transport in pipe networks starting from the non-isothermal Euler equations with friction and heat exchange with the surroundings. An appropriate rescaling of the equations accounting for the large friction, large heat transfer, and low Mach regime leads to simplified isothermal models in the limit epsilon=0. We propose a fully discrete approximation of the isothermal Euler equations using a mixed finite element approach. Based on a reformulation of the equations and relative energy estimates, we derive convergence estimates that hold uniformly in the scaling to a parabolic gas model. We finally extend some ideas and results also to the non-isothermal regime
A Powerful Robust Cubic Hermite Collocation Method for the Numerical Calculations and Simulations of the Equal Width Wave Equation
In this article, non-linear Equal Width-Wave (EW) equation will be
numerically solved . For this aim, the non-linear term in the equation is
firstly linearized by Rubin-Graves type approach. After that, to reduce the
equation into a solvable discretized linear algebraic equation system which is
the essential part of this study, the Crank-Nicolson type approximation and
cubic Hermite collocation method are respectively applied to obtain the
integration in the temporal and spatial domain directions. To be able to
illustrate the validity and accuracy of the proposed method, six test model
problems that is single solitary wave, the interaction of two solitary waves,
the interaction of three solitary waves, the Maxwellian initial condition,
undular bore and finally soliton collision will be taken into consideration and
solved. Since only the single solitary wave has an analytical solution among
these solitary waves, the error norms Linf and L2 are computed and compared to
a few of the previous works available in the literature. Furthermore, the
widely used three invariants I1, I2 and I3 of the proposed problems during the
simulations are computed and presented. Beside those, the relative changes in
those invariants are presented. Also, a comparison of the error norms Linf and
L2 and these invariants obviously shows that the proposed scheme produces
better and compatible results than most of the previous works using the same
parameters. Finally, von Neumann analysis has shown that the present scheme is
unconditionally stable.Comment: 25 pages, 9 tables, 6 figure
Drift-diffusion models for innovative semiconductor devices and their numerical solution
We present charge transport models for novel semiconductor devices which may include ionic species as well as their thermodynamically consistent finite volume discretization
Nonlinear dimensionality reduction then and now: AIMs for dissipative PDEs in the ML era
This study presents a collection of purely data-driven workflows for
constructing reduced-order models (ROMs) for distributed dynamical systems. The
ROMs we focus on, are data-assisted models inspired by, and templated upon, the
theory of Approximate Inertial Manifolds (AIMs); the particular motivation is
the so-called post-processing Galerkin method of Garcia-Archilla, Novo and
Titi. Its applicability can be extended: the need for accurate truncated
Galerkin projections and for deriving closed-formed corrections can be
circumvented using machine learning tools. When the right latent variables are
not a priori known, we illustrate how autoencoders as well as Diffusion Maps (a
manifold learning scheme) can be used to discover good sets of latent variables
and test their explainability. The proposed methodology can express the ROMs in
terms of (a) theoretical (Fourier coefficients), (b) linear data-driven (POD
modes) and/or (c) nonlinear data-driven (Diffusion Maps) coordinates. Both
Black-Box and (theoretically-informed and data-corrected) Gray-Box models are
described; the necessity for the latter arises when truncated Galerkin
projections are so inaccurate as to not be amenable to post-processing. We use
the Chafee-Infante reaction-diffusion and the Kuramoto-Sivashinsky dissipative
partial differential equations to illustrate and successfully test the overall
framework.Comment: 27 pages, 22 figure
ADI schemes for heat equations with irregular boundaries and interfaces in 3D with applications
In this paper, efficient alternating direction implicit (ADI) schemes are
proposed to solve three-dimensional heat equations with irregular boundaries
and interfaces. Starting from the well-known Douglas-Gunn ADI scheme, a
modified ADI scheme is constructed to mitigate the issue of accuracy loss in
solving problems with time-dependent boundary conditions. The unconditional
stability of the new ADI scheme is also rigorously proven with the Fourier
analysis. Then, by combining the ADI schemes with a 1D kernel-free boundary
integral (KFBI) method, KFBI-ADI schemes are developed to solve the heat
equation with irregular boundaries. In 1D sub-problems of the KFBI-ADI schemes,
the KFBI discretization takes advantage of the Cartesian grid and preserves the
structure of the coefficient matrix so that the fast Thomas algorithm can be
applied to solve the linear system efficiently. Second-order accuracy and
unconditional stability of the KFBI-ADI schemes are verified through several
numerical tests for both the heat equation and a reaction-diffusion equation.
For the Stefan problem, which is a free boundary problem of the heat equation,
a level set method is incorporated into the ADI method to capture the
time-dependent interface. Numerical examples for simulating 3D dendritic
solidification phenomenons are also presented
A computational multi-scale approach for brittle materials
Materials of industrial interest often show a complex microstructure which directly influences their macroscopic material behavior. For simulations on the component scale, multi-scale methods may exploit this microstructural information. This work is devoted to a multi-scale approach for brittle materials. Based on a homogenization result for free discontinuity problems, we present FFT-based methods to compute the effective crack energy of heterogeneous materials with complex microstructures
Operator Learning Enhanced Physics-informed Neural Networks for Solving Partial Differential Equations Characterized by Sharp Solutions
Physics-informed Neural Networks (PINNs) have been shown as a promising
approach for solving both forward and inverse problems of partial differential
equations (PDEs). Meanwhile, the neural operator approach, including methods
such as Deep Operator Network (DeepONet) and Fourier neural operator (FNO), has
been introduced and extensively employed in approximating solution of PDEs.
Nevertheless, to solve problems consisting of sharp solutions poses a
significant challenge when employing these two approaches. To address this
issue, we propose in this work a novel framework termed Operator Learning
Enhanced Physics-informed Neural Networks (OL-PINN). Initially, we utilize
DeepONet to learn the solution operator for a set of smooth problems relevant
to the PDEs characterized by sharp solutions. Subsequently, we integrate the
pre-trained DeepONet with PINN to resolve the target sharp solution problem. We
showcase the efficacy of OL-PINN by successfully addressing various problems,
such as the nonlinear diffusion-reaction equation, the Burgers equation and the
incompressible Navier-Stokes equation at high Reynolds number. Compared with
the vanilla PINN, the proposed method requires only a small number of residual
points to achieve a strong generalization capability. Moreover, it
substantially enhances accuracy, while also ensuring a robust training process.
Furthermore, OL-PINN inherits the advantage of PINN for solving inverse
problems. To this end, we apply the OL-PINN approach for solving problems with
only partial boundary conditions, which usually cannot be solved by the
classical numerical methods, showing its capacity in solving ill-posed problems
and consequently more complex inverse problems.Comment: Preprint submitted to Elsevie
Very High-Order A-stable Stiffly Accurate Diagonally Implicit Runge-Kutta Methods with Error Estimators
A numerical search approach is used to design high-order diagonally implicit
Runge-Kutta (DIRK) schemes equipped with embedded error estimators, some of
which have identical diagonal elements (SDIRK) and explicit first stage
(ESDIRK). In each of these classes, we present new A-stable schemes of order
six (the highest order of previously known A-stable DIRK-type schemes) up to
order eight. For each order, we include one scheme that is only A-stable as
well as schemes that are L-stable, stiffly accurate, and/or have stage order
two. The latter types require more stages, but give better convergence rates
for differential-algebraic equations (DAEs), and those which have stage order
two give better accuracy for moderately stiff problems. The development of the
eighth-order schemes requires, in addition to imposing A-stability, finding
highly accurate numerical solutions for a system of 200 equations in over 100
variables, which is accomplished via a combination of global and local
optimization strategies. The accuracy, stability, and adaptive stepsize control
of the schemes are demonstrated on diverse problems
Hyperboloidal discontinuous time-symmetric numerical algorithm with higher order jumps for gravitational self-force computations in the time domain
Within the next decade the Laser Interferometer Space Antenna (LISA) is due
to be launched, providing the opportunity to extract physics from stellar
objects and systems, such as \textit{Extreme Mass Ratio Inspirals}, (EMRIs)
otherwise undetectable to ground based interferometers and Pulsar Timing Arrays
(PTA). Unlike previous sources detected by the currently available
observational methods, these sources can \textit{only} be simulated using an
accurate computation of the gravitational self-force. Whereas the field has
seen outstanding progress in the frequency domain, metric reconstruction and
self-force calculations are still an open challenge in the time domain. Such
computations would not only further corroborate frequency domain calculations
and models, but also allow for full self-consistent evolution of the orbit
under the effect of the self-force. Given we have \textit{a priori} information
about the local structure of the discontinuity at the particle, we will show
how to construct discontinuous spatial and temporal discretisations by
operating on discontinuous Lagrange and Hermite interpolation formulae and
hence recover higher order accuracy. In this work we demonstrate how this
technique in conjunction with well-suited gauge choice (hyperboloidal slicing)
and numerical (discontinuous collocation with time symmetric) methods can
provide a relatively simple method of lines numerical algorithm to the problem.
This is the first of a series of papers studying the behaviour of a
point-particle prescribing circular geodesic motion in Schwarzschild in the
\textit{time domain}. In this work we describe the numerical machinery
necessary for these computations and show not only our work is capable of
highly accurate flux radiation measurements but it also shows suitability for
evaluation of the necessary field and it's derivatives at the particle limit
Parameter-uniformly convergent numerical scheme for singularly perturbed delay parabolic differential equation via extended B-spline collocation
This paper presents a parameter-uniform numerical method to solve the time dependent singularly perturbed delay parabolic convection-diffusion problems. The solution to these problems displays a parabolic boundary layer if the perturbation parameter approaches zero. The retarded argument of the delay term made to coincide with a mesh point and the resulting singularly perturbed delay parabolic convection-diffusion problem is approximated using the implicit Euler method in temporal direction and extended cubic B-spline collocation in spatial orientation by introducing artificial viscosity both on uniform mesh. The proposed method is shown to be parameter uniform convergent, unconditionally stable, and linear order of accuracy. Furthermore, the obtained numerical results agreed with the theoretical results
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