A high-order quasi-conservative discontinuous Galerkin (DG) method is proposed for the numerical simulation of compressible multi-component flows. A distinct feature of the method is a predictor-corrector strategy to define the grid velocity. A Lagrangian mesh is first computed based on the flow velocity and then used as an initial mesh in a moving mesh method (the moving mesh partial differential equation or MMPDE method ) to improve its quality. The fluid dynamic equations are discretized in the direct arbitrary Lagrangian-Eulerian framework using DG elements and the non-oscillatory kinetic flux while the species equation is discretized using a quasi-conservative DG scheme to avoid numerical oscillations near material interfaces. A selection of one- and two-dimensional examples are presented to verify the convergence order and the constant-pressure-velocity preservation property of the method. They also demonstrate that the incorporation of the Lagrangian meshing with the MMPDE moving mesh method works well to concentrate mesh points in regions of shocks and material interfaces.
In this paper, a high order quasi-conservative discontinuous Galerkin (DG) method using the non-oscillatory kinetic flux is proposed for the 5-equation model of compressible multi-component flows with Mie-Gruneisen equation of state. The method mainly consists of three steps: firstly, the DG method with the non-oscillatory kinetic flux is used to solve the conservative equations of the model; secondly, inspired by Abgralls idea, we derive a DG scheme for the volume fraction equation which can avoid the unphysical oscillations near the material interfaces; finally, a multi-resolution WENO limiter and a maximum-principle-satisfying limiter are employed to ensure oscillation-free near the discontinuities, and preserve the physical bounds for the volume fraction, respectively. Numerical tests show that the method can achieve high order for smooth solutions and keep non-oscillatory at discontinuities. Moreover, the velocity and pressure are oscillation-free at the interface and the volume fraction can stay in the interval [0,1].
This work settles the problem of constructing entropy stable non-oscillatory (ESNO) fluxes by framing it as a least square optimization problem. A flux sign stability condition is introduced and utilized to construct arbitrary order entropy stable flux as a convex combination of entropy conservative and non-oscillatory flux. This simple approach is robust which does not explicitly requires the computation of costly dissipation operator and high order reconstruction of scaled entropy variable for constructing the diffusion term. The numerical diffusion is optimized in the sense that entropy stable flux reduces to the underlying non-oscillatory flux. Different non-oscillatory entropy stable fluxes are constructed and used to compute the numerical solution of various standard scalar and systems test problems. Computational results show that entropy stable schemes are comparable in term of non-oscillatory nature of schemes using only the underlying non-oscillatory fluxes. Moreover, these entropy stable schemes maintains the formal order of accuracy of the lower order flux used in the convex combination.
In this study, a novel physics-data-driven Bayesian method named Heat Conduction Equation assisted Bayesian Neural Network (HCE-BNN) is proposed. The HCE-BNN is constructed based on the Bayesian neural network, it is a physics-informed machine learning strategy. Compared with the existed pure data driven method, to acquire physical consistency and better performance of the data-driven model, the heat conduction equation is embedded into the loss function of the HCE-BNN as a regularization term. Hence, the proposed method can build a more reliable model by physical constraints with less data. The HCE-BNN can handle the forward and inverse problems consistently, that is, to infer unknown responses from known partial responses, or to identify boundary conditions or material parameters from known responses. Compared with the exact results, the test results demonstrate that the proposed method can be applied to both heat conduction forward and inverse problems successfully. In addition, the proposed method can be implemented with the noisy data and gives the corresponding uncertainty quantification for the solutions.
For a class of fourth order gradient flow problems, integration of the scalar auxiliary variable (SAV) time discretization with the penalty-free discontinuous Galerkin (DG) spatial discretization leads to SAV-DG schemes. These schemes are linear and shown unconditionally energy stable. But the reduced linear systems are rather expensive to solve due to the dense coefficient matrices. In this paper, we provide a procedure to pre-evaluate the auxiliary variable in the piecewise polynomial space. As a result, the computational complexity of $O(mathcal{N}^2)$ reduces to $O(mathcal{N})$ when exploiting the conjugate gradient (CG) solver. This hybrid SAV-DG method is more efficient and able to deliver satisfactory results of high accuracy. This was also compared with solving the full augmented system of the SAV-DG schemes.
The micro-macro (mM) decomposition approach is considered for the numerical solution of the Vlasov--Poisson--Lenard--Bernstein (VPLB) system, which is relevant for plasma physics applications. In the mM approach, the kinetic distribution function is decomposed as $f=mathcal{E}[boldsymbol{rho}_{f}]+g$, where $mathcal{E}$ is a local equilibrium distribution, depending on the macroscopic moments $boldsymbol{rho}_{f}=int_{mathbb{R}}boldsymbol{e} fdv=langleboldsymbol{e} frangle_{mathbb{R}}$, where $boldsymbol{e}=(1,v,frac{1}{2}v^{2})^{rm{T}}$, and $g$, the microscopic distribution, is defined such that $langleboldsymbol{e} grangle_{mathbb{R}}=0$. We aim to design numerical methods for the mM decomposition of the VPLB system, which consists of coupled equations for $boldsymbol{rho}_{f}$ and $g$. To this end, we use the discontinuous Galerkin (DG) method for phase-space discretization, and implicit-explicit (IMEX) time integration, where the phase-space advection terms are integrated explicitly and the collision operator is integrated implicitly. We give special consideration to ensure that the resulting mM method maintains the $langleboldsymbol{e} grangle_{mathbb{R}}=0$ constraint, which may be necessary for obtaining (i) satisfactory results in the collision dominated regime with coarse velocity resolution, and (ii) unambiguous conservation properties. The constraint-preserving property is achieved through a consistent discretization of the equations governing the micro and macro components. We present numerical results that demonstrate the performance of the mM method. The mM method is also compared against a corresponding DG-IMEX method solving directly for $f$.
Dongmi Luo
,Shiyi Li
,Weizhang Huang
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(2021)
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"A quasi-conservative DG-ALE method for multi-component flows using the non-oscillatory kinetic flux"
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Dongmi Luo
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