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We consider quantum graphs with transparent branching points. To design such networks, the concept of transparent boundary conditions is applied to the derivation of the vertex boundary conditions for the linear Schrodinger equation on metric graphs. This allows to derive simple constraints, which make usual Kirchhoff-type boundary conditions at the vertex equivalent to the transparent ones. The approach is applied to quantum star and tree graphs. However, extension to more complicated graph topologies is rather straightforward.

In this paper we consider the valuation of swing options with the possibility of incorporating spikes in the underlying electricity price. This kind of contracts are modelled as path dependent options with multiple exercise rights. From the mathematical point of view the valuation of these products is posed as a sequence of free boundary problems where two consecutive exercise rights are separated by a time period. Due to the presence of jumps, the complementarity problems are associated with a partial-integro differential operator. In order to solve the pricing problem, we propose appropriate numerical methods based on a Crank–Nicolson semi-Lagrangian method for the time discretization of the differential part of the operator, jointly with the explicit treatment of the integral term by using the Adams–Bashforth scheme and combined with biquadratic Lagrange finite elements for space discretization. In addition, we use an augmented Lagrangian active set method to cope with the early exercise feature. Moreover, we employ appropriate artificial boundary conditions to treat the unbounded domain numerically. Finally, we present some numerical results in order to illustrate the proper behaviour of the numerical schemes.

We consider the reflectionless transport of sine-Gordon solitons on a line. Transparent boundary conditions for the sine-Gordon equation on a line are derived using the so-called potential approach. Our numerical implementation of these novel boundary conditions proves the absence of backscattering in the transmission of sine-Gordon solitons through the boundary of the finite domains under consideration.

In this paper we enhance the well-known fifth order WENO shock-capturing scheme by using deep learning techniques. This fine-tuning of an existing algorithm is implemented by training a rather small neural network to modify the smoothness indicators of the WENO scheme in order to improve the numerical results especially at discontinuities. In our approach no further post-processing is needed to ensure the consistency of the method, which simplifies the method and increases the effect of the neural network. Moreover, the convergence of the resulting scheme can be theoretically proven. We demonstrate our findings with the inviscid Burgers' equation, the Buckley-Leverett equation and the 1-D Euler equations of gas dynamics. Hereby we investigate the classical Sod problem and the Lax problem and show that our novel method outperforms the classical fifth order WENO schemes in simulations where the numerical solution is too diffusive or tends to overshoot at shocks., Comment: 31 pages 12 figures, 6 tables

The concept of the iterative parabolic approximation based on the multiscale technique is discussed. This approach is compared with the traditional ways to derive the wide-angle parabolic equation. While the latter fail in the nonlinear case, the multiscale derivation technique leading to iterative parabolic equations can be easily adapted to handle it. The nonlinear iterative parabolic approximations for the wave propagation in Kerr media are presented. An example demonstrating the capability of iterative parabolic equations to take nonparaxial propagation effects in Kerr media into account is considered.

Recently a new approach to the construction of high-order parabolic approximations for the Helmholtz equation was developed. These approximations have the form of the system of iterative parabolic equations, where the solution of the n-th equation is used as an input term for the ( n + 1 ) -th equation. In this study the transparent boundary conditions for such systems of coupled parabolic equations are derived. The existence and uniqueness of the solution of the initial boundary value problem for the system of iterative parabolic equations with the derived boundary conditions are proved. The well-posedness of this problem is also established and an unconditionally stable finite difference scheme for its solution is proposed.

In this article we combine the ideas of high-order (HO) and alternating direction implicit (ADI) schemes on sparse grids for diffusion equations with mixed derivatives. With the help of HO and ADI schemes solutions can be computed, which are fourth-order accurate in space and second-order accurate in time. For each implicit step of the ADI scheme we use a high-order-compact (HOC) discretisation such that the computational effort consists of only solving tridiagonal systems. In order to reduce the number of grid points, we use the combination technique to construct a solution defined on the sparse grid. This approach allows to further reduce the computational effort and memory consumption.

In this work we present a result on the non-existence of monotone, consistent linear discrete approximation of order higher than 2. This is an essential ingredient, if we want to solve numerically nonlinear and particularly Hamilton–Jacobi–Bellman (HJB) equations.

The modelling of sound propagation in the ocean by the solution of mode parabolic equations is discussed. Mode parabolic equations can be obtained as the one-way approximation to horizontal refraction equations for modal amplitudes. Their wide-angle capabilities depend on the order of the Pade approximation of the involved pseudo-differential operators. Various aspects of numerical solution methods for wide-angle mode parabolic equations are considered in detail, including artificial domain truncation and Cauchy initial data for the point source field approximation. The capabilities of the discussed numerical approaches are demonstrated in several important test cases, including the problems of sound propagation in a penetrable wedge and in a sea with an underwater canyon.

This article deals with artificial boundaries which you encounter when a large spatial domain is confined to a smaller computational domain. Such an artificial boundary condition should not preferably interact with the fluid at all. Standard boundary conditions, e.g., a pressure or velocity condition, result in unphysical reflections. So far, existing artificial boundary conditions for the lattice Boltzmann method (LBM) are transferred from macroscopic formulations.In this work we propose novel discrete artificial boundary conditions (ABCs) which are tailored on the LBM's mesoscopic level. They are derived directly for the chosen LBM with the aim of higher accuracy. We describe the idea of discrete ABCs in a three velocity (D1Q3) model governing the Navier-Stokes equations in one dimension. Numerical results finally demonstrate the superiority of our new boundary condition in terms of accuracy compared to previously used ABCs.

Mayfield’s numerical implementation of transparent boundary condition for the Schrodinger-type parabolic equations is revisited. An inaccuracy in the original proof of the conditional stability for the resulting scheme is pointed out. The highly unusual and impressive original result is reestablished and a new proof is presented. Some further remarks and estimates on the instability which occurs when the Mayfield condition is violated are given.

Force-gradient decomposition methods are used to improve the energy preservation of symplectic schemes applied to Hamiltonian systems. If the potential is composed of different parts with strongly varying dynamics, this multirate potential can be exploited by coupling force-gradient decomposition methods with splitting techniques for multi-time scale problems to further increase the accuracy of the scheme and reduce the computational costs. In this paper, we derive novel force-gradient nested methods and test them numerically. We apply them on the three-body problem, modified for a better observation of the advantageous properties, needed for the future research.

When using a lattice Boltzmann method on an unbounded (or very large) domain one has to confine this spatial domain to a computational domain. This is realized by introducing so-called artificial boundary conditions. Until recently, characteristic boundary conditions for the Euler equations were considered and adapted to the lattice Boltzmann method. In this work we propose novel discrete artificial boundary conditions which are derived directly for the chosen lattice Boltzmann model, i.e., on the discrete level. They represent the first exact artificial boundary conditions for lattice Boltzmann methods. Doing so, we avoid any detour of considering continuous equations and obtain boundary conditions that are perfectly adapted to the chosen numerical scheme. We illustrate the idea for a one dimensional, two velocity (D1Q2) lattice Boltzmann method and show how the computational efficiency can be increased by a finite memory approach. Analytical investigations and numerical results finally demonstrate the advantages of our new boundary condition compared to previously used artificial boundary conditions.

For numerically solving fluid dynamics problems efficiently one is often facing the problem of having to confine the computational domain to a small domain of interest introducing so-called non-reflecting boundary conditions (NRBCs). In this work we address the problem of supplying NRBCs in fluid simulations in two space dimensions using the lattice Boltzmann method (LBM): so-called characteristic boundary conditions are revisited and transferred to the framework of lattice Boltzmann simulations. Numerical tests show clearly that the unwanted unphysical reflections can be reduced significantly by applying our newly developed methods. Hereby the key idea is to transfer and generalize Thompson's boundary conditions originally developed for the nonlinear Euler equations of gas dynamics to the setting of lattice Boltzmann methods. Finally, we give strong numerical evidence that the proposed methods possess a long-time stability property.

In this paper we will derive an SIR model describing vaccination as well as waning immunity and propose a finite difference scheme for its solution together with some qualitative results. For the modeling of the waning immunity we assume a statistical distribution for the level of antibodies depending on the time lapsed since individual's full recovery or vaccination. We arrive at a system of two ODEs and two PDEs that we reduce to a model of just two ODEs and a few algebraic equations. Next, we propose and implement an efficient numerical scheme to solve this reduced model, based on finite differences. To illustrate our findings we provide graphical results and discuss some qualitative properties of the solutions. Additionally, we derive formulas for the basic reproduction number R 0 and the effective reproduction number R ( t ) of the reduced model and show the behavior of solutions for examples with R 0 > 1 and R 0 1 .

In this note we derive, using the subequation method, a new nonstandard finite difference scheme (NSFD) for a class of convection-diffusion equations having constant coefficients. Despite the fact that this scheme has nonlinear denominator functions of the step sizes (even for linear PDEs), it has a couple of favourable properties: it is explicit and due to its construction it reproduces important properties of the solution of the parabolic PDE. This proposed method conserves, by construction, the positivity of the solution if one choses a right combination of spatial and temporal step sizes and hence it is perfectly suited for solving for example air pollution problems or the Black-Scholes equation for the valuation of standard options, since it avoids negative values for the calculated prices. Finally, we illustrate the usefulness of this newly proposed method on a classical benchmark example from the literature.

Recently, Galapon [E.A. Galapon, Delta-convergent sequences that vanish at the support of the limit Dirac delta function, J. Phys. A 42 (2009) 175–201] has posed the question of the existence of delta-convergent sequences that vanish at the support of the limit Dirac delta function and gave an example of sequences of this type. It is a sequence of even functions that do not have a compact support. Motivated by the question, in this short note we develop some results concerning delta sequences and show more examples of delta sequences of the type with or without compact support and that are even or not even.

Nonlinear Black–Scholes equations have been increasingly attracting interest over the last two decades, since they provide more accurate values by taking into account more realistic assumptions, such as transaction costs, risks from an unprotected portfolio, large investor’s preferences or illiquid markets (which may have an impact on the stock price), the volatility, the drift and the option price itself.In this paper we will focus on several models from the most relevant class of nonlinear Black–Scholes equations for European and American options with a volatility depending on different factors, such as the stock price, the time, the option price and its derivatives due to transaction costs. We will analytically approach the option price by transforming the problem for a European Call option into a convection-diffusion equation with a nonlinear term and the free boundary problem for an American Call option into a fully nonlinear nonlocal parabolic equation defined on a fixed domain following Ševčovič’s idea. Finally, we will present the results of different numerical discretization schemes for European options for various volatility models including the Leland model, the Barles and Soner model and the Risk adjusted pricing methodology model.

In this work we construct and analyze discrete artificial boundary conditions (ABCs) for different finite difference schemes to solve nonlinear Schrodinger equations. These new discrete boundary conditions are motivated by the continuous ABCs recently obtained by the potential strategy of Szeftel. Since these new nonlinear ABCs are based on the discrete ABCs for the linear problem we first review the well-known results for the linear Schrodinger equation. We present our approach for a couple of finite difference schemes, including the Crank-Nicholson scheme, the Duran-Sanz-Serna scheme, the DuFort-Frankel method and several split-step (fractional-step) methods such as the Lie splitting, the Strang splitting and the relaxation scheme of Besse. Finally, several numerical tests illustrate the accuracy and stability of our new discrete approach for the considered finite difference schemes.

Transparent boundary conditions (TBCs) are an important tool for the truncation of the computational domain in order to compute solutions on an unbounded domain. In this work we want to show how the standard assumption of 'compactly supported data' could be relaxed and derive TBCs for a generalized Schrodinger equation directly for the numerical scheme on the discrete level. With this inhomogeneous TBCs it is not necessary that the initial data lies completely inside the computational region. However, an increased computational effort must be accepted.

We present a new discrete Adomian decomposition method to approximate the theoretical solution of discrete nonlinear Schrodinger equations. The method is examined for plane waves and for single soliton waves in case of continuous, semi-discrete and fully discrete Schrodinger equations. Several illustrative examples and Mathematica program codes are presented.

This paper deals with the efficient numerical solution of the two-dimensional one-way Helmholtz equation posed on an unbounded domain. In this case, one has to introduce artificial boundary conditions to confine the computational domain. The main topic of this work is the construction of the so-called discrete transparent boundary conditions for state-of-the-art parabolic equation methods, namely a split-step discretization of the high-order parabolic approximation and the split-step Padé algorithm of Collins. Finally, several numerical examples arising in optics and underwater acoustics illustrate the efficiency and accuracy of our approach.

In this work we deal with the numerical solution of some problems of air pollution. Since the problems are posed on unbounded domains we have to introduce artificial boundaries to confine the computational region. We construct and analyse (discrete) transparent boundary conditions for an implicit difference scheme. We discuss the concepts of positivity and monotonicity of difference schemes and briefly consider these properties of difference schemes for advection-diffusion equations arising in problems of air (and water) pollution. The efficiency and accuracy of our method is illustrated by an example.

In this work we construct and analyse transparent boundary conditions (TBCs) for general systems of parabolic equations. These TBCs are constructed for the fully discrete scheme (@q-method, finite differences), in order to maintain unconditional stability of the scheme and to avoid numerical reflections. The discrete transparent boundary conditions (DTBCs) are discrete convolutions in time and are constructed using the solution of the Z-transformed exterior problem. We will analyse the numerical error of these convolution coefficients caused by the inverse Z-transformation. Since the DTBCs are non-local in time and thus very costly to evaluate, we present approximate DTBCs of a sum-of-exponentials form that allow for a fast calculation of the boundary terms. Finally, we will use our approximate DTBCs for an example of a fluid stochastic Petri net and present numerical results.

This paper is concerned with transparent boundary conditions (TBCs) for wide angle “parabolic” equations (WAPEs) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs introduce slight numerical reflections at this artificial boundary and also render the overall Crank?Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC is derived from the fully discretized whole-space problem that is reflection-free and yields an unconditionally stable scheme. While we shall assume a uniform discretization in range, the interior depth discretization (i.e. in the water column) may be nonuniform, and we shall discuss strategies for the “best exterior discretization” (i.e. in the sea bottom). The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems. In the literature different WAPEs (or WAPE and the standard “parabolic” equation) have been coupled in the water and the sea bottom. We analyze under which conditions this yields a hybrid model that is conservative for the acoustic field.