A new shock-tracking technique that avoids re-meshing the computational grid around the moving shock-front was recently proposed by the authors [1]. This paper describes further algorithmic improvements which make the extrapolated Discontinuity Tracking Technique (eDIT) capable of dealing with complex shock-topologies featuring shock-shock and shock-wall interactions. Various test-cases are included to describe the key features of the methodology and prove its order-of-convergence properties.
Traditional probabilistic methods for the simulation of advection-diffusion equations (ADEs) often overlook the entropic contribution of the discretization, e.g., the number of particles, within associated numerical methods. Many times, the gain in a
ccuracy of a highly discretized numerical model is outweighed by its associated computational costs or the noise within the data. We address the question of how many particles are needed in a simulation to best approximate and estimate parameters in one-dimensional advective-diffusive transport. To do so, we use the well-known Akaike Information Criterion (AIC) and a recently-developed correction called the Computational Information Criterion (COMIC) to guide the model selection process. Random-walk and mass-transfer particle tracking methods are employed to solve the model equations at various levels of discretization. Numerical results demonstrate that the COMIC provides an optimal number of particles that can describe a more efficient model in terms of parameter estimation and model prediction compared to the model selected by the AIC even when the data is sparse or noisy, the sampling volume is not uniform throughout the physical domain, or the error distribution of the data is non-IID Gaussian.
There is a need to accurately simulate materials with complex electromagnetic properties when modelling Ground Penetrating Radar (GPR), as many objects encountered with GPR contain water, e.g. soils, curing concrete, and water-filled pipes. One of wi
dely-used open-source software that simulates electromagnetic wave propagation is gprMax. It uses Yees algorithm to solve Maxwells equations with the Finite-Difference Time-Domain (FDTD) method. A significant drawback of the FDTD method is the limited ability to model materials with dispersive properties, currently narrowed to specific set of relaxation mechanisms, namely multi-Debye, Drude and Lorentz media. Consequently, modelling any arbitrary complex material should be done by approximating it as a combination of these functions. This paper describes work carried out as part of the Google Summer of Code (GSoC) programme 2021 to develop a new module within gprMax that can be used to simulate complex dispersive materials using multi-Debye expansions in an automatic manner. The module is capable of modelling Havriliak-Negami, Cole-Cole, Cole-Davidson, Jonscher, Complex-Refractive Index Models, and indeed any arbitrary dispersive material with real and imaginary permittivity specified by the user.
We build a multi-element variant of the smoothness increasing accuracy conserving (SIAC) shock capturing technique proposed for single element spectral methods by Wissink et al. (B.W. Wissink, G.B. Jacobs, J.K. Ryan, W.S. Don, and E.T.A. van der Weid
e. Shock regularization with smoothness-increasing accuracy-conserving Dirac-delta polynomial kernels. Journal of Scientific Computing, 77:579--596, 2018). In particular, the baseline scheme of our method is the nodal discontinuous Galerkin spectral element method (DGSEM) for approximating the solution of systems of conservation laws. It is well known that high-order methods generate spurious oscillations near discontinuities which can develop in the solution for nonlinear problems, even when the initial data is smooth. We propose a novel multi-element SIAC filtering technique applied to the DGSEM as a shock capturing method. We design the SIAC filtering such that the numerical scheme remains high-order accurate and that the shock capturing is applied adaptively throughout the domain. The shock capturing method is derived for general systems of conservation laws. We apply the novel SIAC filter to the two-dimensional Euler and ideal magnetohydrodynamics (MHD) equations to several standard test problems with a variety of boundary conditions.
A series of shock capturing schemes based on nonuniform nonlinear weighted interpolation on nonuniform points are developed for conservation laws. Smoothness indicator and discrete conservation laws are discussed. To make fair comparisons between dif
ferent types of schemes, the properties of eigenvalues of spatial discretization matrices are proved. And the proposed schemes are compared with Weighted Compact Nonlinear Schemes (WCNS) and Flux Reconstruction or Correction Procedure via Reconstruction (FR/CPR) in dispersion, dissipation properties and numerical accuracy. Then, the proposed shock capturing schemes are used as subcell limiters for high-order FR/CPR and the hybrid scheme has superiority in data transformation and satisfying discrete conservation laws. Accuracy, discrete conservation laws and shock capturing properties are tested. Numerical results in one and two dimensions are provided to illustrate that the proposed schemes have good properties in shock capturing and can be applied as subcell limiters for FR/CPR.
In this paper, we study a multi-scale deep neural network (MscaleDNN) as a meshless numerical method for computing oscillatory Stokes flows in complex domains. The MscaleDNN employs a multi-scale structure in the design of its DNN using radial scalin
gs to convert the approximation of high frequency components of the highly oscillatory Stokes solution to one of lower frequencies. The MscaleDNN solution to the Stokes problem is obtained by minimizing a loss function in terms of L2 normof the residual of the Stokes equation. Three forms of loss functions are investigated based on vorticity-velocity-pressure, velocity-stress-pressure, and velocity-gradient of velocity-pressure formulations of the Stokes equation. We first conduct a systematic study of the MscaleDNN methods with various loss functions on the Kovasznay flow in comparison with normal fully connected DNNs. Then, Stokes flows with highly oscillatory solutions in a 2-D domain with six randomly placed holes are simulated by the MscaleDNN. The results show that MscaleDNN has faster convergence and consistent error decays in the simulation of Kovasznay flow for all four tested loss functions. More importantly, the MscaleDNN is capable of learning highly oscillatory solutions when the normal DNNs fail to converge.