Finite Difference Methods,Theory and Applications

We study two finite element computational models for solving coupled problems involving flow in a fracture and flow in poroelastic media. The Brinkman equation is used in the fracture, while the Biot system of poroelasticity is employed in the surrounding media. Appropriate equilibrium and kinematic conditions are imposed on the interfaces. We focus on the approximation of the interface conditions, which in this context feature the interaction of different variables, such as velocities, displacements, stresses and pressures. The aim of this study is to compare the Lagrange multiplier and the Nitsche’s methods applied to enforce these non standard interface conditions.

This work deals with the derivation of a novel transparent boundary condition (TBC) for the coupling of the standard “parabolic” equation (SPE) in underwater acoustics (assuming cylindrical symmetry) with an elastic parabolic equation (EPE) for modelling the sea bottom extending hereby the existing TBCs for a fluid model of the seabed.

In the present paper the well-posedness of the elliptic differential equation

$$\begin{aligned} -u^{\prime \prime }(t)+Au(t)=f(t)(-\infty <t<\infty ) \end{aligned}$$

-

u

″

(

t

)

+

A

u

(

t

)

=

f

(

t

)

(

-

∞

<

t

<

∞

)

in an arbitrary Banach space

E

with the general positive operator in Hö lder spaces

$$C^{\beta }(\mathbb {R},E_{\alpha })$$

C

β

(

R

,

E

α

)

is established. The exact estimates in Hölder norms for the solution of the problem for elliptic equations are obtained. The high order of accuracy two-step difference schemes generated by an exact difference scheme or by Taylor’s decomposition on three points for the approximate solutions of this differential equation are studied. The well-posedness of the these difference schemes in the difference analogy of Hölder spaces

$$C^{\beta }(\mathbb {R}_{\tau }, E_{\alpha })$$

C

β

(

R

τ

,

E

α

)

are obtained. The almost coercive inequality for solutions in

$$C(\mathbb {R}_{\tau },E)$$

C

(

R

τ

,

E

)

of these difference schemes is established.

The paper serves as a review on the basic results showing how functional analytic tools have been applied in numerical analysis. It deals with abstract Cauchy problems and present how their solutions are approximated by using space and time discretisations. To this end we introduce and apply the basic notions of operator semigroup theory. The convergence is analysed through the famous theorems of Trotter and Kato, Lax, and Chernoff. We also list some of their most important applications.

In problems of mathematical physics, Trefftz approximations by definition involve functions that satisfy the differential equation of the problem. The power and versatility of such approximations is illustrated with an overview of a number of application areas: (i) finite difference Trefftz schemes of arbitrarily high order; (ii) boundary difference Trefftz methods analogous to boundary integral equations but completely singularity-free; (iii) Discontinuous Galerkin (DG) Trefftz methods for Maxwell’s electrodynamics; (iv) numerical and analytical nonreflecting Trefftz boundary conditions; (v) non-asymptotic homogenization of electromagnetic and photonic metamaterials.

In this paper we present a further development of our asymptotic comparison principle, applying it for some new important classes of initial boundary value problem for the nonlinear singularly perturbed time periodic parabolic equations, which are called in applications as reaction-diffusion-advection equations. We illustrate our approach for the new problem with balanced nonlinearity. The theorems, which states the existence of the periodic solution with internal layer, gives it’s asymptotic approximation and state their Lyapunov stability are proved.

For boundary value problems with singularity, we developed the theory of finite difference schemes based on concept of an

$$R_\nu $$

R

ν

-generalized solution. The difference scheme is constructed, the rate of convergence of the approximate solution to the

$$R_\nu $$

R

ν

-generalized solution in the norm of the Sobolev weighted space is established.

This paper deals with the calculation of linear and quadratic functionals of approximate solutions obtained by the finite element method. It is shown that under certain conditions the output functionals of an approximate solution are computed with higher order of accuracy than that of the solution itself. These abstract results are illustrated by two numerical examples for the Poisson equation.

This work deals with the problem of choosing a time step for the numerical solution of boundary value problems for parabolic equations. The problem solution is derived using the fully implicit scheme, whereas a time step is selected via explicit calculations. Using the explicit scheme, we calculate the solution at a new time level. We employ this solution in order to obtain the solution at the previous time level (the implicit scheme, explicit calculations). This solution should be close to the solution of our problem at this time level with a prescribed accuracy. Such an algorithm leads to explicit formulas for the calculation of the time step and takes into account both the dynamics of the problem solution and changes in coefficients of the equation and in its right-hand side.

This paper provides a brief survey on some of the recent numerical techniques and schemes for solving Hamilton-Jacobi-Bellman equations arising in pricing various options. These include optimization methods in both infinite and finite dimensions and discretization schemes for nonlinear parabolic PDEs.

Recently, considerable attention has been drawn to stochastic controlled systems with hidden Markov chains.

We deal with an initial-boundary value problem for the generalized time-dependent Schrödinger equation with variable coefficients in an unbounded

n

–dimensional parallelepiped (

$$n\ge 1$$

n

≥

1

). To solve it, the Crank-Nicolson in time and the polylinear finite element in space method with the discrete transparent boundary conditions is considered. We present its stability properties and derive new error estimates

$$O(\tau ^2+|h|^2)$$

O

(

τ

2

+

|

h

|

2

)

uniformly in time in

$$L^2$$

L

2

space norm, for

$$n\ge 1$$

n

≥

1

, and mesh

$$H^1$$

H

1

space norm, for

$$1\le n\le 3$$

1

≤

n

≤

3

(a superconvergence result), under the Sobolev-type assumptions on the initial function. Such estimates are proved for methods with the discrete TBCs for the first time.

The initial-boundary value problem for the delay parabolic partial differential equation with nonlocal conditions is studied. The convergence estimates for solutions of first and second order of accuracy difference schemes in Hölder norms are obtained. The theoretical statements are supported by a numerical example.

A method of investigation of numerical schemes deriving from the variational formulation of the problem (variational- difference method and FEM) is discusses. The method is based on the reduction of the numerical schemes to the canonical finite difference form. The resulting numerical scheme standard notation in the form of a grid operator equality is used for analyzing its approximation, stability and other properties. The application of this approach to a wider classes of finite elements (from the simplest ones to the Hermitian elements and serendipities) is discussed. These opportunities are illustrated by the analysis of FEM schemes for Timoshenko shells and elasticity dynamic problems.

The numerical method for solving of the hypersonic nonequilibrium aerogasdynamics problems is suggested. The method is based on the full three-dimensional Navier-Stokes equations, supplemented by the equations of chemical kinetics and the finite difference TVD method. The developed algorithms are implemented in the computer-aided software package SIGMA. The results of simulation of the hypersonic flow about the spherical nose segment of a model hypersonic vehicle are presented.

This paper describes an application of the surface harmonics method to derivation of few-group finite difference equations for neutron flux distribution in a 3D triangular-lattice reactor model. The Boltzmann neutron transport equation is used as the original equation. Few-group finite difference equations are derived, which describe the neutron importance distribution (the multiplication factor in the homogeneous eigenvalue problem) in the reactor core. The derived finite difference equations remain adjoint to each other like the original equation of neutron transport and its adjoint equation. Non-diffusion approximations apply to calculation of a whole reactor core if we increase the number of trial functions for describing the neutron flux distribution in each cell and the size of the matrices of the few-group coefficients for finite difference equations.

In this work we present so-called generalized Picard iterations (GPI) – a family of iterative processes which allows to solve mildly stiff ODE systems using implicit Runge–Kutta (IRK) methods without storing and inverting Jacobi matrices. The key idea is to solve nonlinear equations arising from the base IRK method by special iterative process based on the idea of artificial time integration. By construction these processes converge for all asymptotically stable linear ODE systems and all A-stable base IRK methods at arbitrary large time steps. The convergence rate is limited by the value of “stiffness ratio”, but not by the value of Lipschitz constant of Jacobian. The computational scheme is well suited for parallelization on systems with shared memory. The presented numerical results exhibit that the proposed GPI methods in case of mildly stiff problems can be more advantageous than traditional explicit RK methods.

In this paper a mathematical model for simulation of thermal fields from wells located in permafrost area is considered, which takes into account basic physical, technological, and climatic factors that lead to a nonlinear boundary condition on the surface of the soil. To find the thermal fields a finite-difference method is used to solve the problem of Stefan type, and solvability of the corresponding difference problem is proved. Possibilities of the developed software are presented to carry out various numerical experiments and make long-term forecasts in simulations of thermal fields in the system “well – permafrost” with annual cycle of thawing/freezing the upper layers of the soil due to seasonal temperature changes, intensity of solar radiation and technical parameters of the wells. Comparison of numerical and experimental data are in good agreement (difference is about 5

$$\%$$

%

) due to, in particular, that the software adapts to the geographic location by using special iterative algorithm of determination of the parameters, included in the non-linear boundary condition on the soil surface.

In this paper we apply modified implicit methods for nonlinear dynamical systems related to constrained and non-separable Hamiltonian problems. The application of well-known standard Runge-Kutta integrator methods based on splitting schemes failed, while the energy conservation is no longer guaranteed. We propose a novel class of iterative implicit method that resolves the nonlinearity and achieve an asymptotic symplectic behavior. In comparison to explicit symplectic methods we achieve more accurate results for 5–10 iterations for only double computational time.

In this paper we present a model to simulate textile as used in pest control. For this application, textile is coated with a repellent, protecting the user from insect bites, and one wants to determine optimal material properties. The model extends an existing 3 scale method to allow for simulations in saturated conditions. This is achieved with the addition of an overlapping domain decomposition approach for the fiber-yarn interaction.

With the model we present how the performance of a coating can be determined: how much material is required, what evaporative properties are needed, how can the coating be replenished? Furthermore, the model can be used to evaluate the effects of the used textile substrate, like the type and number of fibers or the weaving structure. Lastly, it can be used to validate simple first-order models of coated textile.

Numerical results indicate the 3 scale approach is valid. The influence of different textile properties on the effectiveness of the resulting textile component is presented.

For solving of the Navier-Stokes equations describing the incompressible viscous fluid flows the Splitting on physical factors Method for Incompressible Fluid flows (SMIF) with hybrid explicit finite difference scheme (second-order accuracy in space, minimum scheme viscosity and dispersion, capable for work in the wide range of Reynolds (

Re

) and internal Froude (

Fr

) numbers and monotonous) based on the Modified Central Difference Scheme (MCDS) and the Modified Upwind Difference Scheme (MUDS) with a special switch condition depending on the velocity sign and the signs of the first and second differences of the transferred functions has been developed and successfully applied. At the present paper the description of the numerical method SMIF and its applications for simulation of the 3D separated homogeneous and density stratified fluid flows around a sphere and a circular cylinder are demonstrated.

In this paper, we consider an inverse problem of determining the time-dependent thermal conductivity from Cauchy data in a one-dimensional heat equation with space-dependent heat capacity. The parabolic partial differential equation is discretised using the finite -difference method and the inverse problem is recast as a nonlinear least-squares minimization. This is solved using the

lsqnonlin

routine from the MATLAB toolbox. Numerical results are presented and discussed showing that accurate and stable numerical solutions are achieved.

This work deals with the inverse problem of simultaneous determination of two unknown time-dependent terms in a one-dimensional parabolic equation. The additional information is given by two integral observations. We prove theorems of existence and uniqueness of solution. We also give estimates of maximum modulus of unknown right-hand side and unknown coefficient of the equation with constants derived explicitly in terms of input data.

In this paper effectiveness of several parallel implementations of the finite element method is investigated for an algorithm of a numerical solution of the boundary function problem for the shallow water equations. The parallel technologies MPI, OpenMP and MPI+OpenMP are used.

We consider an unconditionally stable splitting scheme for solving coupled systems of equations arising in poroelasticity and thermoelasticity problems. The scheme is based on splitting the systems of equation into physical processes, which means the transition to the new time level is associated with solving separate sub-problems for displacement and pressure/temperature. The stability of the scheme is achieved by switching to three-level finite-difference scheme with weight. We present stability estimates of the scheme based on Samarskii’s theory of stability for operator-difference schemes. We provide numerical experiments supporting the stability estimates of the splitting scheme.

In this paper we present a new mathematical model describing acquired immune response to viral infection. The model is formulated as a system of six ordinary differential equations (ODE). Conditions for existence, uniqueness and non-negativity of the solutions are studied. Numerical simulations for the case of dominating cellular immunity and various initial values of concentrations of virus particles are presented and discussed.

In this paper we present numerical methods for solving a non-linear time-fractional parabolic model. To cope with non-local in time nature of the problem, we exploit the idea of the two-grid method and develop fast numerical algorithms. Moreover, we show that suitable modifications of the standard two-grid technique lead to significant reduction of the computational time. Numerical results are also discussed.

A family of

four level conservative finite difference schemes

(FDS) for the multidimensional Boussinesq Equation is constructed and studied theoretically. A preservation of the discrete energy for this approach is established. We prove that the discrete solution of the FDS converges to the exact solution with a second order of convergence with respect to space and time mesh steps in the first discrete Sobolev norm and in the uniform norm. The numerical experiments for the one-dimensional problem confirm the theoretical rate of convergence and the preservation of the discrete energy in time.

In this paper, we deal with the following

p

-Laplacian fractional boundary value problem:

$$ \phi _p(D_{0+}^\alpha u(t))+f(t,u(t))=0,~0<t<1$$

ϕ

p

(

D

0

+

α

u

(

t

)

)

+

f

(

t

,

u

(

t

)

)

=

0

,

0

<

t

<

1

,

$$u(0)=u'(0)=u'(1)=0, \ $$

u

(

0

)

=

u

′

(

0

)

=

u

′

(

1

)

=

0

,

where

$$2<\alpha \leqslant 3$$

2

<

α

⩽

3

is a real number.

$$D_{0+}^\alpha $$

D

0

+

α

is the standard Riemann–Liouville differentiation, and

$$f:[0,1]\times [0,+\infty )\rightarrow [0,+\infty )$$

f

:

[

0

,

1

]

×

[

0

,

+

∞

)

→

[

0

,

+

∞

)

is continuous. By the properties of the Green function and some fixed-point theorems on cone, some existence and multiplicity results of positive solutions are obtained. As applications, examples are presented to illustrate the main results.

The new computational model for the seismic wave propagation is proposed, the governing equations of which are written in terms of velocities, stress tensor and small rotation of element of the medium. The properties of wavefields in the prestressed medium are studied and some examples showing anisotropy of prestressed state are discussed. The staggered grid numerical method is developed for solving the governing equations of the model and numerical examples are presented.

A numerical-analytical algorithm for modeling of seismic and acoustic-gravity waves propagation is applied to a heterogeneous “Earth-Atmosphere" model. Seismic wave propagation in an elastic half-space is described by a system of first-order dynamic equations of elasticity theory. The propagation of acoustic-gravity waves in the atmosphere is described by the linearized Navier-Stokes equations with the wind. The algorithm is based on the integral Laguerre transform with respect to time, the finite integral Fourier transform with respect to a spatial coordinate combined with a finite difference method for the reduced problem.

An extrapolation algorithm is considered for solving linear fractional differential equations in this paper, which is based on the direct discretization of the fractional differential operator. Numerical results show that the approximate solutions of this numerical method has the expected asymptotic expansions.

Finite difference methods for solving two-sided space-fractional partial differential equations are studied. The space-fractional derivatives are the left-handed and right-handed Riemann-Liouville fractional derivatives which are expressed by using Hadamard finite-part integrals. The Hadamard finite-part integrals are approximated by using piecewise quadratic interpolation polynomials and a numerical approximation scheme of the space-fractional derivative with convergence order

$$O(\varDelta x^{3- \alpha }), \, 1<\alpha <2$$

O

(

Δ

x

3

-

α

)

,

1

<

α

<

2

is obtained. A shifted implicit finite difference method is introduced for solving two-sided space-fractional partial differential equation and we prove that the order of convergence of the finite difference method is

$$O (\varDelta t + \varDelta x^{\min (3- \alpha , \beta )}), 1< \alpha <2, \, \beta >0$$

O

(

Δ

t

+

Δ

x

min

(

3

-

α

,

β

)

)

,

1

<

α

<

2

,

β

>

0

, where

$$\varDelta t, \varDelta x$$

Δ

t

,

Δ

x

denote the time and space step sizes, respectively. Numerical examples are presented and compared with the exact analytical solution for its order of convergence.

We consider a class of boundary value problems for fractional integro-differential equations. Using an integral equation reformulation of the boundary value problem, we first study the regularity of the exact solution. Based on the obtained regularity properties and spline collocation techniques, the numerical solution of the boundary value problem by suitable non-polynomial approximations is discussed. Optimal global convergence estimates are derived and a super-convergence result for a special choice of grid and collocation parameters is given. A numerical illustration is also presented.

We present a rational spectral collocation method for pricing American vanilla and butterfly spread options. Due to the early exercise possibilities, free boundary conditions are associated with both of these PDEs. The problem is first reformulated as a variational inequality. Then, by adding a penalty term, the resulting variational inequality is transformed into a nonlinear advection-diffusion-reaction equation on fixed boundaries. This nonlinear PDE is discretised in asset (space) direction by means of rational interpolation using suitable barycentric weights and transformed Chebyshev points. This gives a system of stiff nonlinear ODEs which is then integrated using an implicit fourth-order Lobatto time-integration method. We carried out extensive comparisons with other results obtained by using some existing methods found in literature and observed that our approach is very competitive.

In this work, the exact solution of the Riemann problem for first-order nonlinear partial equation with non-convex state function in

$$Q_T=\{(x,t)|x\in I=\left( -\infty ,\ \ \infty \right) ,\ t\in \left[ 0,T\right) \}\subset R^2$$

Q

T

=

{

(

x

,

t

)

|

x

∈

I

=

-

∞

,

∞

,

t

∈

0

,

T

}

⊂

R

2

is found. Here

$$F\in C^2{(Q}_T)\ $$

F

∈

C

2

(

Q

T

)

and

$$\ F^{''}(u)$$

F

′

′

(

u

)

change their signs, that is

F

(

u

) has convex and concave parts. In particular, the state function

$$F\left( u\right) =-{\cos u\ }$$

F

u

=

-

cos

u

on

$$\ \left[ \frac{\pi }{2},\frac{3\pi }{2}\right] $$

π

2

,

3

π

2

and

$$\ \left[ \frac{\pi }{2},\frac{5\pi }{2}\right] $$

π

2

,

5

π

2

is discussed. For this, when it is necessary, the auxiliary problem which is equivalent to the main problem is introduced. The solution of the proposed problem permits constructing the weak solution of the main problem that conserves the entropy condition. In some cases, depending on the nature of the investigated problem a convex or a concave hull is constructed. Thus, the exact solutions are found by using these functions.

Based on the equations of the dynamics of a piecewise-homogeneous elastic material, parallel computational algorithms are developed to simulate the process of stress and strain wave propagation in a medium consisting of a large number of blocks, interacting through compliant interlayers. For the description of waves in a block medium with thick interlayers the orthotropic couple-stress continuum theory, taking into account the symmetry of elastic properties relative to the coordinate planes, can be applied. By comparing the elastic wave velocities in the framework of piecewise-homogeneous model and continuum model, the simple method is obtained to estimate mechanical parameters of an orthotropic Cosserat continuum, modeling a block medium. In two-dimensional formulation of the orthotropic model, the computational algorithm and the program system are worked out for the analysis of propagation of elastic waves. The comparison showed good qualitative agreement between the results of computations of waves, caused by localized impulses, by the model of a block medium with compliant interlayers and the model of an orthotropic Cosserat continuum.

Numerical algorithm for solving dynamic problems of the theory of viscoelastic medium of Kelvin–Voigt is worked out on the basis of Ivanov’s method of constructing finite difference schemes with prescribed dissipative properties. In one-dimensional problem the results of computations are compared with the exact solution, describing the propagation of plane monochromatic waves. When solving two-dimensional problems, the total approximation method based on the splitting of the system with respect to the spatial variables is applied. The algorithm is tested on solving the problem of traveling surface waves. For illustration of the method, the numerical solution of Lamb’s problem about instantaneous action of concentrated force on the boundary of a half-plane is represented in viscoelastic formulation.

This work is devoted to finding a solution of Riemann problem for the first order nonlinear partial equation which describes the traffic flow on highway. When

$$\rho _{\ell }>\rho _r$$

ρ

ℓ

>

ρ

r

, the solution is presented as a piecewise continuous function, where

$$\rho _{\ell }$$

ρ

ℓ

and

$$\rho _r$$

ρ

r

are the densities of cars on the left and right side of the intersection respectively. On the contrary case, a shock of which the location is unknown beforehand arises in the solution. In this case, a special auxiliary problem is introduced, the solution of which makes it possible to write the exact solution showing the locations of shock. For the realization of the proposed method, the parameters of the flow are also found.

In this work we consider the coupled linear system of equations for temperature and displacements which describes the thermoelastic behaviour of the body. For numerical solution we approximate our system using finite element method. As model problem for simulation we consider the thermomechanical state of the ceramic substrates with metallization, which are used for the manufacturing of light-emitting diode modules. The results of numerical simulation of the 3D problem in the complex geometric area are presented.

This paper presents a fast direct solver for 3D discretized linear systems using the supernodal multifrontal method together with low-rank approximations. For linear systems arising from certain partial differential equations (PDEs) such as elliptic equations, during the Gaussian elimination of the matrices with Nested Dissection ordering, the fill-in of L and U factors loses its sparsity and contains dense blocks with low-rank property. Off-diagonal blocks can be efficiently approximated with low-rank matrices; diagonal blocks approximated with semiseparable structures called hierarchically semiseparable (HSS) representations. Matrix operations in the multifrontal method are performed in low-rank arithmetic. We present efficient way to organize the HSS structured operations along the elimination. To compress dense blocks into low-rank or HSS structures, we use effective cross approximation (CA) approach. We also use idea of adaptive balancing between robust arithmetic for computing the small dense blocks and low-rank matrix operations for handling with compressed ones while performing the Gaussian elimination. This new proposed solver can be essentially parallelized both on architecture with shared and distributed memory and can be used as effective preconditioner. To check efficient of our solver we compare it with Intel MKL PARDISO - the high performance direct solver. Memory and performance tests demonstrate up to 3 times performance and memory gain for the 3D problems with more than

$$10^6$$

10

6

unknowns. Therefore, proposed multifrontal HSS solver can solve large problems, which cannot be resolved by direct solvers because of large memory consumptions.

The article describes the rare mesh scheme based on finite element method, describes the methods of constructing such schemes are described arrangements of nodes, describes methods of calculation tasks based on rare mesh schemes, the problem of static, Numerical solutions of different tasks based on rare mesh scheme circuit compares the results with the known systems.

Among inverse problems for PDEs we distinguish coefficient inverse problems, which are associated with the identification of the right-hand side of an equation using some additional information. When considering time-dependent problems, the identification of the right-hand side dependences on space and on time is usually separated into individual problems. We have linear inverse problems; this situation essentially simplify their study. This work deals with the problem of determining in a multidimensional parabolic equation the right-hand side that depends on time only. To solve numerically a inverse problem we use standard finite difference approximations in space. The computational algorithm is based on a special decomposition, where the transition to a new time level is implemented via solving two standard elliptic problems.

An open geothermal system consisting of injection and productive wells is considered. Hot water from production well is used and became cooler, and injection well returns the cold water into the aquifer. To simulate this open geothermal system a three–dimensional nonstationar mathematical model of the geothermal system is developed taking into account the most important physical and technical parameters of the wells to describe processes of heat transfer and thermal water filtration in a aquifer. Results of numerical calculations, which, in particular, are used to determine an optimal parameters for a geothermal system in North Caucasus, are presented. For example, a distance in the productive layer between the point of hot water inflow and of cold water injection point is considered.

We present an 1D numerical model of heat, steam, and water transfer across a wall consisting of several layers of different materials. The model is the system of coupled diffusion equations for wall temperature; vapor pressure, and water concentration in material pores, with account of vapor condensation and water evaporation. The system of nonlinear PDEs is solved numerically using the finite difference method. The main objective of modeling is simulation of long-term behavior of building wall moisture distribution under influence of seasonal variations in atmospheric air temperature and humidity.

A singularly perturbed initial-boundary value problem for a parabolic equation known in applications as the reaction-diffusion equation is considered. An asymptotic expansion of the solution with moving front is constructed. Using the asymptotic method of differential inequalities we prove the existence and estimate the asymptotic expansion for such solutions. The method is based on well-known comparison theorems and formal asymptotics for the construction of upper and lower solutions in singularly perturbed problems with internal and boundary layers.

A priori estimates for a new approach to constructing vector splitting schemes in mixed FEM for heat transfer problems are presented. Heat transfer problem is considered in the mixed weak formulation approximated by Raviart-Thomas finite elements of lowest order on rectangular meshes. The main idea of the considered approach is to develop splitting schemes for the heat flux using well-known splitting scheme for the scalar function of flux divergence. Based on flux decomposition into discrete divergence-free and potential (orthogonal) components, a priori estimates for 2D and 3D vector splitting schemes are presented. Special attention is given to the additional smoothness requirements imposed on the initial heat flux. The role of these requirements is illustrated by several numerical examples.

Interpolation formulas for the functions of one variable with a boundary layer component are investigated. An interpolated function corresponds to a solution of a singular perturbed problem. An application of Lagrange interpolation on a uniform mesh leads to significant errors. Two approaches for a interpolation of a function with a boundary layer component are considered: a fitting of the interpolation formula to a boundary layer component and the application of Lagrange interpolation on Shishkin mesh. Numerical results are discussed.