The ghost fluid method (GFM) provides a simple way to simulate the interaction of immiscible materials. Especially, the modified GFM (MGFM) and its variants, based on the solutions of multi-material Riemann problems, are capable of faithfully taking into account the effects of nonlinear wave interaction and material property near the interface. Reasonable treatments for ghost fluid states or interface conditions have been shown to be crucial when applied to various interfacial phenomena involving large discontinuity and strong nonlinearity. These methods, therefore, have great potential in engineering applications. In this paper, we review the development of such methods. The methodologies of representative GFM-based algorithms for definition of interface conditions are illustrated and compared to each other. The research progresses in design principle and accuracy analysis are briefly described. Some steps and techniques for multi-dimensional extension are also summarized. In addition, we present some progresses in more challenging scientific problems, including a variety of fluid/solid-fluid/solid interactions with complex physical properties. Of course the challenges faced by researchers in this field are also discussed.

In this contribution we study the formal ability of a multi-resolution-times lattice Boltzmann scheme to approximate isothermal and thermal compressible Navier Stokes equations with a single particle distribution. More precisely, we consider a total of 12 classical square lattice Boltzmann schemes with prescribed sets of conserved and nonconserved moments. The question is to determine the algebraic expressions of the equilibrium functions for the nonconserved moments and the relaxation parameters associated to each scheme. We compare the fluid equations and the result of the Taylor expansion method at second order accuracy for bidimensional examples with a maximum of 17 velocities and three-dimensional schemes with at most 33 velocities. In some cases, it is not possible to fit exactly the physical model. For several examples, we adjust the Navier Stokes equations and propose nontrivial expressions for the equilibria.

Due to its highly oscillating solution, the Helmholtz equation is numerically challenging to solve. To obtain a reasonable solution, a mesh size that is much smaller than the reciprocal of the wavenumber is typically required (known as the pollution effect). High-order schemes are desirable, because they are better in mitigating the pollution effect. In this paper, we present a high-order compact finite difference method for 2D Helmholtz equations with singular sources, which can also handle any possible combinations of boundary conditions (Dirichlet, Neumann, and impedance) on a rectangular domain. Our method is sixth-order consistent for a constant wavenumber, and fifth-order consistent for a piecewise constant wavenumber. To reduce the pollution effect, we propose a new pollution minimization strategy that is based on the average truncation error of plane waves. Our numerical experiments demonstrate the superiority of our proposed finite difference scheme with reduced pollution effect to several state-of-the-art finite difference schemes, particularly in the critical pre-asymptotic region where $kh$ is near 1 with $k$ being the wavenumber and $h$ the mesh size.

We propose finite-volume schemes for the Cahn-Hilliard equation which unconditionally and discretely preserve the boundedness of the phase field and the dissipation of the free energy. Our numerical framework is applicable to a variety of free-energy potentials, including Ginzburg-Landau and Flory-Huggins, to general wetting boundary conditions, and to degenerate mobilities. Its central thrust is the upwind methodology, which we combine with a semi-implicit formulation for the free-energy terms based on the classical convex-splitting approach. The extension of the schemes to an arbitrary number of dimensions is straightforward thanks to their dimensionally split nature, which allows to efficiently solve higher-dimensional problems with a simple parallelisation. The numerical schemes are validated and tested through a variety of examples, in different dimensions, and with various contact angles between droplets and substrates.

The article opens a series of publications devoted to a systematic study of numerical errors behind the shock wave when using high-order Godunov-type schemes, including in combination with the artificial viscosity approach. The proposed paper describes the numerical methods used in the study, and identifies the main factors affecting the accuracy of the solution for the case of one-dimensional gas dynamic problems. The physical interpretation of the identified factors is given and their influence on the grid convergence is analyzed.

The traditional stochastic homogenization method can obtain homogenized solutions of elliptic problems with stationary random coefficients. However, many random composite materials in scientific and engineering computing do not satisfy the stationary assumption. To overcome the difficulty, we propose a normalizing field flow induced two-stage stochastic homogenization method to efficiently solve the random elliptic problem with non-stationary coefficients. By applying the two-stage stochastic homogenization method, the original elliptic equation with random and fast oscillatory coefficients is approximated as an equivalent elliptic equation, where the equivalent coefficients are obtained by solving a set of cell problems. Without the stationary assumption, the number of cell problems is large and the corresponding computational cost is high. To improve the efficiency, we apply the normalizing field flow model to learn a reference Gaussian field for the random equivalent coefficients based on a small amount of data, which is obtained by solving the cell problems with the finite element method. Numerical results demonstrate that the newly proposed method is efficient and accurate in tackling high dimensional partial differential equations in composite materials with complex random microstructures.

In this paper, we give the first rigorous error estimation of the Weak Adversarial Neural Networks (WAN) in solving the second order parabolic PDEs. By decomposing the error into approximation error and statistical error, we first show the weak solution can be approximated by the $ReLU^2$ with arbitrary accuracy, then prove that the statistical error can also be efficiently bounded by the Rademacher complexity of the network functions, which can be further bounded by some integral related with the covering numbers and pseudo-dimension of $ReLU^2$ space. Finally, by combining the two bounds, we prove that the error of the WAN method can be well controlled if the depth and width of the neural network as well as the sample numbers have been properly selected. Our result also reveals some kind of freedom in choosing sample numbers on $∂Ω$ and in the time axis.

The industry-standard constrained pressure residual (CPR) algorithm is often able to effectively improve the robustness behavior and the convergence speed of linear iterations for isothermal reservoir simulation. In this paper, we present and study an improved extension of CPR to the constrained pressure-temperature residual (CPTR) version for non-isothermal reservoir problems in heterogeneous porous media. In the proposed preconditioner, the corresponding approximations for the inverse of matrices are computed under a domain decomposition framework by using the restricted additive Schwarz (RAS) algorithm, to equally deal with the coupled thermal-pressure-saturation reservoir system and highly exploit the parallelism of supercomputer platforms. Moreover, we introduce and develop a family of multilevel CPTR preconditioners with suitable coarse grid corrections, to further improve the applicability of this two-stage preconditioner for large-scale computation. Numerical results for strong heterogeneous flow problems show that the new approach can dramatically improve the convergence of linear iterations, and demonstrate the superiority of CPTR over the commonly used RAS preconditioners. The parallel scalability of the non-isothermal reservoir simulator is also studied versus a supercomputer with tens of thousands of processors.