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High-order methods have attracted considerable attention in the CFD fields because of their potential in achieving higher accuracy with a lower cost than low order methods. In this paper, a new class of high order spectral volume method based on Chebyshev polynomials as an approximation function called Chebyshev Spectral Volume (CSV) method is presented for hyperbolic conservation laws in quadrilateral...
A surface moving mesh method is presented for general surfaces with or without explicit parameterization. The method can be viewed as a nontrivial extension of the moving mesh partial differential equation method that has been developed for bulk meshes and demonstrated to work well for various applications. The main challenges in the development of surface mesh movement come from the fact that the...
Uncertainty Quantification for nonlinear hyperbolic problems becomes a challenging task in the vicinity of shocks. Standard intrusive methods, such as Stochastic Galerkin (SG), lead to oscillatory solutions and can result in non-hyperbolic moment systems. The intrusive polynomial moment (IPM) method guarantees hyperbolicity but comes at higher numerical costs. In this paper, we filter the generalized...
In this paper, we study the analytical properties and the numerical methods for the Bogoliubov-de Gennes equations (BdGEs) describing the elementary excitation of Bose-Einstein condensates around the mean field ground state, which is governed by the Gross-Pitaevskii equation (GPE). Derived analytical properties of BdGEs can serve as benchmark tests for numerical algorithms and three numerical methods...
We present a nodal Discontinuous Galerkin (DG) scheme for the Cahn–Hilliard equation that satisfies the summation–by–parts simultaneous–approximation–term (SBP–SAT) property. The latter permits us to show that the discrete free–energy is bounded, and as a result, the scheme is provably stable. The scheme and the stability proof are presented for general curvilinear three–dimensional hexahedral meshes...
Multiscale modeling is a systematic approach to describe the behavior of complex systems by coupling models from different scales. The approach has been demonstrated to be very effective in areas of science as diverse as materials science, climate modeling and chemistry. However, routine use of multiscale simulations is often hindered by the very high cost of individual at-scale models. Approaches...
A fractional step method for solving the steady-state, incompressible, Navier-Stokes equations is presented. The proposed iterative method uses an estimate for the optimal under-relaxation coefficient at each iteration. This estimate uses prior information that is already available in the iterative method so the calculation has negligible cost. Numerical tests show that the convergence rate of the...
Global stability modes of flows provide significant insight into their dynamics. Direct methods to obtain these modes are restricted by the daunting sizes and complexity of Jacobians encountered in general three-dimensional flows. Jacobian-free iterative approaches such as those based on Arnoldi algorithm have greatly alleviated the required computational burden by considering a smaller subspace formed...
We introduce a high-order weight-adjusted discontinuous Galerkin (WADG) scheme for the numerical solution of three-dimensional (3D) wave propagation problems in anisotropic porous media. We use a coupled first-order symmetric stress-velocity formulation [1,2]. Careful attention is directed at (a) the derivation of an energy-stable penalty-based numerical flux, which offers high-order accuracy in presence...
Convergence failure and slow convergence rates are among the biggest challenges with solving the system of non-linear equations numerically. Although mitigated, such issues still linger when using strictly small time steps and unconditionally stable fully implicit schemes. The price that comes with restricting time steps to small scales is the enormous computational load, especially in large-scale...
When the discontinuous Galerkin (DG) method is applied to hyperbolic problems in two dimensions on triangular meshes and paired with an explicit time integration scheme, an exact CFL condition is not known. The stability condition which is most usually implemented involves scaling the time step by the smallest radius of the inscribed circle in every cell. However, this is known to not provide a tight...
The goal of this article is to make automatic data assimilation for a landslide tsunami model, given by the coupling between a non-hydrostatic multi-layer shallow-water and a Savage-Hutter granular landslide model for submarine avalanches. The coupled model is discretized using a positivity preserving second-order path-conservative finite volume scheme. Then, the data assimilation problem is posed...
In recent years, deep learning has proven to be a viable methodology for surrogate modeling and uncertainty quantification for a vast number of physical systems. However, in their traditional form, such models can require a large amount of training data. This is of particular importance for various engineering and scientific applications where data may be extremely expensive to obtain. To overcome...
The present paper proposes that reconstruction scheme and interpolation scheme can be converted into each other through two series of adapter schemes, which include reconstruction-to-interpolation (RI) adapter schemes and interpolation-to-reconstruction (IR) adapter schemes. For the high-order spatial discretization of the compressible Navier-Stokes equations, the RI adapter schemes can be used to...
Continuum Sensitivity Analysis (CSA) provides an analytic method of computing derivatives for structures, fluids and fluid-structure-interaction problems with respect to shape or value parameters. CSA does not require mesh sensitivity to calculate local shape sensitivity, thereby contributing to its computational efficiency. CSA involves solving linear sensitivity equations with the corresponding...
We study symplectic numerical integration of mechanical systems with a Hamiltonian specified in non-canonical coordinates and its application to guiding-center motion of charged plasma particles in magnetic confinement devices. The technique combines time-stepping in canonical coordinates with quadrature in non-canonical coordinates and is applicable in systems where a global transformation to canonical...
In this article, we investigate computationally some controllability properties of a physical system consisting of three inductively coupled Josephson junctions. This system is modeled by nonlinear ordinary differential equations. A particular attention is given to the optimal control of the transition between equilibrium states, possibly unstable. After defining the control problem cost function,...
A natural medium for wave propagation comprises a coupled bounded heterogeneous region and an unbounded homogeneous free-space. Frequency-domain wave propagation models in the medium, such as the variable coefficient Helmholtz equation, include a faraway decay radiation condition (RC). It is desirable to develop algorithms that incorporate the full physics of the heterogeneous and unbounded medium...
The Vlasov–Maxwell equations are used for the kinetic description of magnetized plasmas. As they are posed in an up to 3+3 dimensional phase space, solving this problem is extremely expensive from a computational point of view. In this paper, we exploit the low-rank structure in the solution of the Vlasov equation. More specifically, we consider the Vlasov–Maxwell system and propose a dynamic low-rank...
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