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We are interested in numerical schemes for the simulation of large scale gas networks. Typical models are based on the isentropic Euler equations with realistic gas constant. The numerical scheme is based on transformation of conservative…
In this paper we present three different numerical approaches to account for curl-type involution constraints in hyperbolic partial differential equations for continuum physics. All approaches have a direct analogy to existing and…
Over the past few decades, there has been substantial interest in evolution equations that involving a fractional-order derivative of order $\alpha\in(0,1)$ in time, due to their many successful applications in engineering, physics, biology…
Finite difference discretization schemes preserving a subgroup of the maximal Lie invariance group of the one-dimensional linear heat equation are determined. These invariant schemes are constructed using the invariantization procedure for…
Fundamental solution of a space fractional convection equation of order $\alpha$ is the probability density function of L\'{e}vy flights with long-tailed $\alpha$-stable jump length distribution. By studying an upwind second-order implicit…
The upwind conservation element and solution element (CESE) scheme is an alternative discontinuity-capturing numerical approach to solving hyperbolic conservation laws. To evaluate the numerical properties of this spatiotemporal coupled…
We propose numerical schemes for the approximate solution of problems defined on the edges of a one-dimensional graph. In particular, we consider linear transport and a drift-diffusion equations, and discretize them by extending Finite…
A numerical algorithm for solving mantle convection problems with strongly variable viscosity is presented. Equations for conservation of mass and momentum for highly viscous and incompressible fluids are solved iteratively by a multigrid…
We prove convergence of the proximal policy gradient method for a class of constrained stochastic control problems with control in both the drift and diffusion of the state process. The problem requires either the running or terminal cost…
We propose two closely--related Lagrangian numerical methods for the simulation of physical processes involving advection, reaction and diffusion. The methods are intended to be used in settings where the flow is nearly incompressible and…
In this paper, a high-order exponential scheme is developed to solve the 1D unsteady convection-diffusion equation with Neumann boundary conditions. The present method applies fourth-order compact exponential difference scheme in spatial…
We consider an optimal control problem on a bounded domain $\Omega\subset\mathbb{R}^2,$ governed by a parabolic convection--diffusion--reaction equation with pointwise control constraints. We follow the optimize--then--discretize approach,…
A new stochastic control problem of a dam-reservoir system installed in a river is analyzed both mathematically and numerically. Water balance dynamics of the reservoir are piece-wise deterministic and are driven by a stochastic…
In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species $c$ and the chemical potential $h$ is $h(c)=\log \frac{c}{1-c}$. We design four different finite…
In this paper, we propose stochastic structure-preserving schemes to compute the effective diffusivity for particles moving in random flows. We first introduce the motion of particles using the Lagrangian formulation, which is modeled by…
In recent years, kinetic equations have been used to model many social phenomena. A key feature of these models is that transition rate kernels involve Dirac delta functions, which capture sudden, discontinuous state changes. Here, we study…
We propose a new numerical approach to compute nonclassical solutions to hyperbolic conservation laws. The class of finite difference schemes presented here is fully conservative and keep nonclassical shock waves as sharp interfaces,…
This paper introduces a numerical approach to solve singularly perturbed convection diffusion boundary value problems for second-order ordinary differential equations that feature a small positive parameter {\epsilon} multiplying the…
This paper presents a new type of genetic algorithm for the set covering problem. It differs from previous evolutionary approaches first because it is an indirect algorithm, i.e. the actual solutions are found by an external decoder…
Developing high-order numerical schemes for two-phase flow in porous media that preserve key physical properties remains a significant challenge in numerical analysis. In this article, we propose a general framework to construct fully…