Treatment of discontinuous and singular source terms in hyperbolic systems

Singular source terms in sub-diffusion equations may lead to the unboundedness of solutions, which will bring a severe reduction of convergence order of existing time-stepping schemes. In this work, we propose two efficient time-stepping schemes for solving sub-diffusion equations with a class of source terms mildly singular in time. One discretization is based on the Gr{\"u}nwald-Letnikov and backward Euler methods. First-order error estimate with respect to time is rigorously established for singular source terms and nonsmooth initial data. The other scheme derived from the second-order backward differentiation formula (BDF) is proved to possess second-order accuracy in time. Further, piecewise linear finite element and lumped mass finite element discretizations in space are applied and analyzed rigorously. Numerical investigations confirm our theoretical results.

Improved FVM for two-layer shallow-water models: Application to the Strait of Gibraltar

A reduced-dimension method for unknown Crank-Nicolson finite element solution coefficient vectors of elastic wave equation with singular source term

Large Time Step Roe scheme for a common 1D two-fluid model

The exact Riemann Solver to the Shallow Water equations for natural channels with bottom steps

In recent years, the application of coupled solver techniques to solve the Navier-Stokes equations has become increasingly popular. The main reason for this, is the increased robustness originating from the implicit and global treatment of the pressure-velocity coupling. The drawback of a coupled solver are the increase in memory requirement and the increased complexity of implementation. However, fully coupled methods are reported to have an overall favorable computational cost when a suitable pre-conditioner and algorithm for solving the resulting set of linear equations are employed. In solving multi-phase flow problems, the coupled solver approach is even more advantageous than in single-phase, due to the presence of large source terms arising from the coupling of the phases. In this chapter, various strategies for the fully coupled approach are discussed. These strategies include employing artificial compressibility, applying physically consistent cell face interpolation, and applying momentum weighted cell face interpolation. The idea behind the strategies is outlined and their advantages and disadvantages are discussed. The treatment of source terms and volume fraction in coupled methods is also shown. Finally, a number of examples of implementations and calculations are presented.

<p>In this paper, we discuss a higher-order convergent numerical method for a two-parameter singularly perturbed differential equation with a discontinuous convection coefficient and a discontinuous source term. The presence of perturbation parameters generates boundary layers, and the discontinuous terms produce interior layers on both sides of the discontinuity. In order to obtain a higher-order convergent solution, a hybrid monotone finite difference scheme is constructed on a piecewise uniform Shishkin mesh, which is adapted inside the boundary and interior layers. On this mesh (including the point of discontinuity), the present method is almost second-order parameter-uniform convergent. The current scheme is compared with the standard upwind scheme, which is used at the point of discontinuity. The numerical experiments based on the proposed scheme show higher-order (almost second-order) accuracy compared to the standard upwind scheme, which provides almost first-order parameter-uniform convergence.</p>

Physics-Informed Neural Networks (PINNs) have emerged as a promising approach for solving acoustic seismic wave equations in geophysical modeling. However, they often encounter challenges such as spectral bias, gradient instability, and slow convergence when applied to complex media, high-frequency wavefields, and singular source terms. To address these limitations, we propose Adaptive SIREN-PINNs, an enhanced framework that integrates learnable amplitude and frequency modulation parameters into the Sinusoidal Representation Networks (SIRENs), which employs periodic activation functions to naturally represent continuous signals and their derivatives, along with a principled weight initialization strategy. This enhancement improves the model’s frequency-aware representation capability, enabling it to better capture both low- and high-frequency components while enhancing gradient propagation stability during training. Furthermore, to mitigate the singularity associated with point sources, we adopt a Gaussian approximation of the Dirac function and reformulate the wave equation in terms of the scattered wavefield, following established regularization practices in seismic wave modeling. Additionally, we conduct a spectral analysis of the Neural Tangent Kernel (NTK) to gain insights into the model’s learning dynamics across different frequency components during training. Numerical experiments conducted on isotropic homogeneous medium, the Marmousi model, and anisotropic TTI/VTI medium demonstrate that the proposed method significantly outperforms conventional PINNs and baseline SIREN-PINNs in terms of wavefield reconstruction accuracy and convergence efficiency. The results validate the effectiveness of the adaptive SIREN mechanism in improving both the expressive power and training stability of PINNs, offering a mesh-free and solver-free alternative to traditional numerical methods.

The Poisson–Boltzmann equation (PBE) is a fundamental implicit solvent continuum model for calculating the electrostatic potential of large ionic solvated biomolecules. However, its numerical solution encounters severe challenges arising from its strong singularity and nonlinearity. In (P. Benner, V. Khoromskaia, B. Khoromskij, C. Kweyu, and M. Stein, “Regularization of Poisson-Boltzmann type equations with singular source terms using the range-separated tensor format,” SIAM J. Sci. Comput., vol. 43, no. 1, pp. A415–A445, 2021; C. Kweyu, V. Khoromskaia, B. Khoromskij, M. Stein, and P. Benner, “Solution decomposition for the nonlinear Poisson-Boltzmann equation using the range-separated tensor format,” arXiv:2109.14073, 2021), the effect of strong singularities was eliminated by applying the range-separated (RS) canonical tensor format (P. Benner, V. Khoromskaia, and B. N. Khoromskij, “Range-separated tensor format for many-particle modeling,” SIAM J. Sci. Comput., vol. 40, no. 2, pp. A1034–A1062, 2018; B. N. Khoromskij, “Range-separated tensor representation of the discretized multidimensional Dirac delta and elliptic operator inverse,” J. Comput. Phys., vol. 401, p. 108998, 2020) to construct a solution decomposition scheme for the PBE. The RS tensor format allows deriving a smooth approximation to the Dirac delta distribution in order to obtain a regularized PBE (RPBE) model. However, solving the RPBE is still computationally demanding due to its high dimension N $\mathcal{N}$ , where N $\mathcal{N}$ is always in the millions. In this study, we propose to apply the reduced basis method (RBM) and the (discrete) empirical interpolation method ((D)EIM) to the RPBE in order to construct a reduced order model (ROM) of low dimension N ≪ N $N\ll \mathcal{N}$ , whose solution accurately approximates the nonlinear RPBE. The long-range potential can be obtained by lifting the ROM solution back to the N $\mathcal{N}$ -space while the short-range potential is directly precomputed analytically, thanks to the RS tensor format. The sum of both provides the total electrostatic potential. The main computational benefit is the avoidance of computing the numerical approximation of the singular electrostatic potential. We demonstrate in the numerical experiments, the accuracy and efficacy of the reduced basis (RB) approximation to the nonlinear RPBE (NRPBE) solution and the corresponding computational savings over the classical nonlinear PBE (NPBE) as well as over the RBM being applied to the classical NPBE.

A diffuse interface–lattice Boltzmann model for conjugate heat transfer with imperfect interface

2D well-balanced augmented ADER schemes for the Shallow Water Equations with bed elevation and extension to the rotating frame

The mixed explicit‐implicit Galerkin finite element method developed previously by the authors is shown to be ideally suited for a wide class of problems arising in subsurface hydrology. These problems include confined saturated flow, unconfined flow under free surface conditions subject to the Dupuit assumption, flow in aquifers which are partly confined and partly unconfined, axisymmetric flow to a well with storage, and flow in saturated‐unsaturated soils. A single computer program, entitled Flump, can now handle all of these problems. The mixed explicit‐implicit solution strategy employed in the program insures a high level of accuracy and computation efficiency in most cases. It eliminates many of the difficulties that groundwater hydrologists have been encountering in trying to simulate extensive aquifer systems by finite elements. Some of the outstanding features of this solution strategy include an automatic control of time step size, reclassification of nodes from explicit to implicit during execution, automatic adjustment of the implicit time‐weighting factor, and the treatment of boundary conditions and source terms as arbitrary functions of time of the state of the system. Five examples are presented to demonstrate the versatility and power of this new approach. A purely physical derivation of the finite element equations which does not rely on the Galerkin formalism is also included in one of the appendices.

Fourier analysis of semi-discrete and space–time stabilized methods for the advective–diffusive–reactive equation: II. SGS

This paper presents new finite volume schemes for hyperbolic conservation system with source terms. The classical finite volume schemes could not accurately simulate the dynamic problems caused by the balance between flux term and source terms. In order to deal with this problem, an approximate Riemann solver with source terms was designed in accordance with the classical HLL approximation Riemann solver. The well-balanced HLL schemes (WB-HLL) was obtained by modifying the flux calculation schemes for one-dimensional Euler equations and ideal MHD with gravity source terms, and a proof for well-balanced property of the new schemes has been presented.Two numerical examples of one-dimensional Euler equation and ideal MHD demonstrated that the WB-HLL scheme has higher accuracy and faster convergence than the classical HLL.

Derivative Riemann solvers for systems of conservation laws and ADER methods