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In this paper, we establish a unified analysis of both the conforming and nonconforming virtual element methods for a class of nonlinear Sobolev equations in which the nonlinear diffusion coefficients, a(u) and b(u), may not satisfy global Lipschitz continuity or elliptical conditions. The discretization for the spatial variables is the virtual element method on general polygonal meshes, whereas the temporal discretization is accomplished with a linearized Crank–Nicolson scheme. By employing a temporal-spatial error splitting argument and introducing a novel projection operator, we prove that the numerical solution is bounded in the L ∞ -norm without any time step-size conditions. Moreover, for both conforming and nonconforming ones, optimal error estimates in L 2 -norm and H 1 -seminorm for the fully scheme are obtained unconditionally. Numerical results are provided on different polygonal meshes to confirm the theoretical analysis. [ABSTRACT FROM AUTHOR]

This work considers the numerical analysis of nonlinear Volterra integrodifferential equations that arise from, e.g., the theory of isotropic viscoelastic rods and plates. Novel properties of the kernel and its derivatives are derived via the Laplace transform, and the regularity of the solutions is proved. Then we apply the Crank–Nicolson method and the second-order convolution quadrature rule to develop the discrete-in-time scheme, and the energy argument is employed to analyze the stability and convergence of the proposed scheme. Numerical experiments are performed to substantiate the theoretical findings. [ABSTRACT FROM AUTHOR]

A constructive numerical approximation of the two-dimensional unsteady stochastic Navier–Stokes equations of an incompressible fluid is proposed via a pseudo-compressibility technique involving a penalty parameter ε . Space and time are discretized through a finite element approximation and an Euler method. The convergence analysis of the suggested numerical scheme is investigated throughout this paper. It is based on a local monotonicity property permitting the convergence toward the unique strong solution of the stochastic Navier–Stokes equations to occur within the originally introduced probability space. [ABSTRACT FROM AUTHOR]

In this work, we construct a fully discrete scheme with finite difference method based on the staggered grids for the viscous Cahn–Hilliard equation. The constructed scheme can satisfy the unconditional dissipation law with original energy. We carry out a rigorous error analysis with superconvergence by introducing an auxiliary function depending on the chemical potential. We obtain second order accuracy in both space and time with l ∞ (0 , T ; H 1 (Ω)) norm for the phase function and l 2 (0 , T ; l 2 (Ω)) for the chemical potential on non-uniform grids. Numerical experiments are presented to verify the theoretical results. [ABSTRACT FROM AUTHOR]

This work focuses on the design of a 2D numerical scheme for the M 1 model on polygonal and conical meshes. This model is nonlinear and approximates the firsts moments of the radiative transfer equation using an entropic closure. Besides, this model admits a diffusion limit as the cross section goes to infinity. It is important for the numerical scheme to be consistent with this limit, that is to say, it should be asymptotic preserving or AP. Such a scheme already exists on polygonal meshes and the present work consists in adapting it to conical meshes. After introducing conical meshes, we explain the construction of the scheme. It is based on an analogy between the M 1 model and the Euler gas dynamics system. We also present a second order reconstruction procedure and we apply it on both polygonal and conical meshes. In the last section, some numerical test cases are given so as to compare the nodal and conical schemes. The limit scheme is studied and we observe numerically that it is consistent with the diffusion equation. [ABSTRACT FROM AUTHOR]

A convection–diffusion problem with a large shift in space is considered. Numerical analysis of high order finite element methods on layer-adapted Durán type meshes, as well as on coarser Durán type meshes in places where weak layers appear, is provided. The theoretical results are confirmed by numerical experiments. [ABSTRACT FROM AUTHOR]

In this paper we conceive and analyze a discrete duality finite volume (DDFV) scheme for the unsteady generalized thermistor problem, including a p-Laplacian for the diffusion and a Joule heating source. As in the continuous setting, the main difficulty in the design of the discrete model comes from the Joule heating term. To cope with this issue, the Joule heating term is replaced with an equivalent key formulation on which a fully implicit scheme is constructed. Introducing a tricky cut-off function in the proposed discretization, we are able to recover the energy estimates on the discrete temperature. Another feature of this approach is that we dispense with the discrete maximum principle on the approximate electric potential, which in essence poses restrictive constraints on the mesh shape. Then, the existence of discrete solution to the coupled scheme is established. Compactness estimates are also shown. Under general assumptions on the data and meshes, the convergence of the numerical scheme is addressed. Numerical results are finally presented to show the efficiency and accuracy of the proposed methodology as well as the behavior of the implemented nonlinear solver. [ABSTRACT FROM AUTHOR]

A rotation-two-component Camassa–Holm (R2CH) system was proposed recently to describe the motion of shallow water waves under the influence of gravity. This is a highly nonlinear and strongly coupled system of partial differential equations. A crucial issue in designing numerical schemes is to preserve invariants as many as possible at the discrete level. In this paper, we present a provable implicit nonlinear difference scheme which preserves at least three discrete conservation invariants: energy, mass, and momentum, and prove the existence of the difference solution via the Browder theorem. The error analysis is based on novel and refined estimates of the bilinear operator in the difference scheme. By skillfully using the energy method, we prove that the difference scheme not only converges unconditionally when the rotational parameter diminishes, but also converges without any step-ratio restriction for the small energy case when the rotational parameter is nonzero. The convergence orders in both settings (zero or nonzero rotation parameter) are O (τ 2 + h 2) for the velocity in the L ∞ -norm and the surface elevation in the L 2 -norm, where τ denotes the temporal stepsize and h the spatial stepsize, respectively. The theoretical predictions are confirmed by a properly designed two-level iteration scheme. Compared with existing numerical methods in the literature, the proposed method demonstrates its effectiveness for long-time simulation over larger domains and superior resolution for both smooth and non-smooth initial values. [ABSTRACT FROM AUTHOR]

By rewriting the Riemann–Liouville fractional derivative as Hadamard finite-part integral and with the help of piecewise quadratic interpolation polynomial approximations, a numerical scheme is developed for approximating the Riemann–Liouville fractional derivative of order α ∈ (1 , 2). The error has the asymptotic expansion (d 3 τ 3 - α + d 4 τ 4 - α + d 5 τ 5 - α + ⋯) + (d 2 ∗ τ 4 + d 3 ∗ τ 6 + d 4 ∗ τ 8 + ⋯) at any fixed time t N = T , N ∈ Z + , where d i , i = 3 , 4 , ... and d i ∗ , i = 2 , 3 , ... denote some suitable constants and τ = T / N denotes the step size. Based on this discretization, a new scheme for approximating the linear fractional differential equation of order α ∈ (1 , 2) is derived and its error is shown to have a similar asymptotic expansion. As a consequence, a high-order scheme for approximating the linear fractional differential equation is obtained by extrapolation. Further, a high-order scheme for approximating a semilinear fractional differential equation is introduced and analyzed. Several numerical experiments are conducted to show that the numerical results are consistent with our theoretical findings. [ABSTRACT FROM AUTHOR]

This article deals with two different numerical approaches for solving singularly perturbed parabolic problems with time delay and interior layers. In both approaches, the implicit Euler scheme is used for the time scale. In the first approach, the upwind scheme is used to deal with the spatial derivatives whereas in the second approach a hybrid scheme is used, comprising the midpoint upwind scheme and the central difference scheme at appropriate domains. Both schemes are applied on two different layer resolving meshes, namely a Shishkin mesh and a Bakhvalov–Shishkin mesh. Stability and error analysis are provided for both schemes. The comparison is made in terms of the maximum absolute errors, rates of convergence, and the computational time required. Numerical outputs are presented in the form of tables and graphs to illustrate the theoretical findings. [ABSTRACT FROM AUTHOR]

We study a weighted alternating direction implicit (ADI) numerical method with variable time steps for two-dimensional diffusion-wave equations. The variable-step Alikhanov formula is employed to approximate the fractional derivatives in an equivalent coupled equations which is generated by the symmetric fractional-order reduction (SFOR) method. By adding a weighted small external term, we obtain a weighted ADI scheme for the diffusion-wave equations. The unconditional stability and convergence are analyzed by energy method, and the optimal temporal convergence order is min { 2 , 3 2 α } , where 1 < α < 2 . The spatial compact scheme combined with the ADI method is also discussed. Numerical examples are provided to confirm the accuracy and efficiency of proposed schemes. [ABSTRACT FROM AUTHOR]

We study in this paper new developments of the Lagrange–Galerkin method for the advection equation. In the first part of the article we present a new improved error estimate of the conventional Lagrange–Galerkin method. In the second part, we introduce a new local projection stabilized Lagrange–Galerkin method, whereas in the third part we introduce and analyze a discontinuity-capturing Lagrange–Galerkin method. Also, attention has been paid to the influence of the quadrature rules on the stability and accuracy of the methods via numerical experiments. [ABSTRACT FROM AUTHOR]

A two-grid temporal second-order scheme for the two-dimensional nonlinear Volterra integro-differential equation with a weakly singular kernel is of concern in this paper. The scheme is targeted to reduce the computation time and to improve the accuracy of the scheme developed by Xu et al. (Appl Numer Math 152:169–184, 2020). The constructed scheme is armed by three steps: First, a small nonlinear system is solved on the coarse grid using a fix-point iteration. Second, Lagrange’s linear interpolation formula is used to arrive at some auxiliary values for the analysis of the fine grid. Finally, a linearized Crank–Nicolson finite difference system is solved on the fine grid. Moreover, the algorithm uses a central difference approximation for the spatial derivatives. In the time direction, the time derivative and integral term are approximated by the Crank–Nicolson technique and product integral rule, respectively. By means of the discrete energy method, stability and space-time second-order convergence of the proposed approach are obtained in L 2 -norm. Finally, the numerical verification is fulfilled as the numerical results of the given numerical experiments agree with the theoretical analysis and verify the effectiveness of the algorithm. [ABSTRACT FROM AUTHOR]

In this article, we propose to construct and analyze a higher order parameter uniform numerical scheme for a class of two dimensional (2D) parabolic singularly perturbed convection–diffusion problem with a multiple boundary turning point. The solution of the considered problem exhibits boundary layers at x = 0 , y = 0 and a corner layer in the neighbourhood of (0, 0). A fractional step method on a uniform mesh is employed in time direction for semi-discretization of the problem. A hybrid scheme on a piecewise uniform mesh is used to discretize the resulting one dimensional (1D) problems in the space variables. Furthermore, Richardson extrapolation is employed in the time variable. The resulting scheme has almost second order of convergence in space direction and second order of convergence in time direction. The theoretical findings are verified through numerical experiments conducted on three test problems. Results are compared with those available in the literature to establish better accuracy and order of convergence of the proposed scheme. [ABSTRACT FROM AUTHOR]

For a backward differentiation formula (BDF) applied to the gradient flow of a semiconvex function, quadratic stability implies gradient stability. Namely, it is possible to build a Lyapunov functional for the discrete-in-time dynamical system, with a restriction on the time step. The maximum time step which can be derived from quadratic stability has previously been obtained for the BDF1, BDF2 and BDF3 schemes. Here, we compute it for the BDF4 and BDF5 methods. We also prove that the BDF6 scheme is not quadratically stable. Our results are based on the tools developed by Dahlquist and other authors to show the equivalence of A-stability and G-stability. We give several applications of gradient stability to the asymptotic behaviour of sequences generated by BDF schemes. [ABSTRACT FROM AUTHOR]

We present an efficient method for the numerical approximation of a general class of two dimensional semilinear parabolic problems on polygonal meshes. The proposed approach takes advantage of the properties of the serendipity version of the Virtual Element Method, which not only reduces the number of degrees of freedom compared to the original Virtual Element Method, but also allows for the introduction of an approximation of the nonlinear term that is computable from the degrees of freedom of the discrete solution with a low computational cost, thus significantly improving the efficiency of the method. An error analysis for the semi-discrete formulation is carried out, and an optimal estimate for the error in the L 2 -norm is obtained. The accuracy and efficiency of the proposed method when combined with a second order Strang operator splitting time discretization is illustrated in our numerical experiments, with approximations up to order 6. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical solution of the two-dimensional Klein-Gordon-Dirac equations by a fourth-order compact finite difference method in space and an energy-preserving Crank-Nicolson-type discretization in time. For convenience of illustrating the conservative properties and investigating the convergence results, we convert the component-wise form of the proposed scheme into an equivalent matrix-vector form. By using the mathematical induction argument and standard energy method, we establish the optimal error estimates under condition τ ≤ 1 | ln (h) | with time step τ and mesh size h. Compared with the condition τ = O (h 1 2) required by existing results in literature for two-dimensional case, this greatly relaxes the dependence of the time step on the grid size. The convergence order of the scalar ϕ and the 2-spinor ψ is of O (τ 2 + h 4) in the maximum norm and the discrete H 1 -norm, respectively. In addition, by using the orthogonal diagonalization technique, a fast solver is designed to solve the proposed scheme. Several numerical results are reported to verify the error estimates and the discrete conservation laws. [ABSTRACT FROM AUTHOR]

We use the scalar auxiliary variable (SAV) reformulation of the nonlinear Schrödinger (NLS) equation to construct structure-preserving SAV–Gauss methods for the NLS equation, namely L 2 -conservative methods satisfying a discrete analogue of the energy (the Hamiltonian) conservation of the equation. This is in contrast to Gauss methods for the standard form of the NLS equation that are L 2 -conservative but not energy-conservative. We also discuss efficient linearizations of the new methods and their conservation properties. [ABSTRACT FROM AUTHOR]

We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a second-order hyperbolic partial differential equation in which, after using the motion equation to eliminate the displacement unknown, the stress tensor remains as the main variable to be found. The resulting variational formulation is shown to be well-posed, and a class of H (div) -conforming semi-discrete schemes is proved to be convergent. Then, we use the Newmark trapezoidal rule to obtain an associated fully discrete scheme, whose main convergence results are also established. Finally, numerical examples illustrating the performance of the method are reported. [ABSTRACT FROM AUTHOR]

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth. [ABSTRACT FROM AUTHOR]

We propose an implicit Newmark method for the time integration of the pressure-stress formulation of a fluid-structure interaction problem. The space Galerkin discretization is based on the Arnold-Falk-Winther mixed finite element method with weak symmetry in the solid and the usual Lagrange finite element method in the acoustic medium. We prove that the resulting fully discrete scheme is well-posed and uniformly stable with respect to the discretization parameters and Poisson ratio, and we provide asymptotic error estimates. Finally, we present numerical tests to confirm the asymptotic error estimates predicted by the theory. [ABSTRACT FROM AUTHOR]

In this paper, we propose a fully discrete mixed finite element method for solving the time-dependent Ginzburg-Landau equations, and prove the convergence of the finite element solutions in general curved polyhedra, possibly nonconvex and multi-connected, without assumptions on the regularity of the solution. Global existence and uniqueness of weak solutions for the PDE problem are also obtained in the meantime. A decoupled time-stepping scheme is introduced, which guarantees that the discrete solution has bounded discrete energy, and the finite element spaces are chosen to be compatible with the nonlinear structure of the equations. Based on the boundedness of the discrete energy, we prove the convergence of the finite element solutions by utilizing a uniform $$L^{3+\delta }$$ regularity of the discrete harmonic vector fields, establishing a discrete Sobolev embedding inequality for the Nédélec finite element space, and introducing a $$\ell ^2(W^{1,3+\delta })$$ estimate for fully discrete solutions of parabolic equations. The numerical example shows that the constructed mixed finite element solution converges to the true solution of the PDE problem in a nonsmooth and multi-connected domain, while the standard Galerkin finite element solution does not converge. [ABSTRACT FROM AUTHOR]

We analyze the discretization of the periodic initial value problem for Kuramoto-Sivashinsky type equations with Burgers nonlinearity by implicit-explicit backward difference formula (BDF) methods, establish stability and derive optimal order error estimates. We also study discretization in space by spectral methods. [ABSTRACT FROM AUTHOR]

This paper is concerned with high-order numerical methods for a class of fractional mobile/immobile convection-diffusion equations. The convection coefficient of the equation may be spatially variable. In order to overcome the difficulty caused by variable coefficient problems, we first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation for the spatial derivative and a second-order difference approximation for the time first derivative and the Caputo time fractional derivative. The local truncation error and the solvability of the resulting scheme are discussed in detail. The (almost) unconditional stability and convergence of the method are proved using a discrete energy analysis method. A Richardson extrapolation algorithm is presented to enhance the temporal accuracy of the computed solution from the second-order to the third-order. Applications using two model problems give numerical results that demonstrate the accuracy of the new method and the high efficiency of the Richardson extrapolation algorithm. [ABSTRACT FROM AUTHOR]

In this work, we develop a new linearized implicit finite volume method for chemotaxis-growth models. First, we derive the scheme for a simplified chemotaxis model arising in embryology. The model consists of two coupled nonlinear PDEs: parabolic convection-diffusion equation with a logistic source term for the cell-density, and an elliptic reaction-diffusion equation for the chemical signal. The numerical approximation makes use of a standard finite volume scheme in space with a special treatment for the convection-diffusion fluxes which are approximated by the classical Il'in fluxes. For the time discretization, we introduce our linearized semi-exponentially fitted scheme. The paper gives a comparison between the proposed scheme and different versions of linearized backward Euler schemes. The existence and uniqueness of a numerical solution to the scheme and its convergence to a weak solution of the studied system are proved. In the last section, we present some numerical tests to show the performance of our method. Our numerical approach is then applied to a chemotaxis-growth model describing bacterial pattern formation. [ABSTRACT FROM AUTHOR]

This paper is concerned with numerical methods for a class of two-dimensional fractional convection-subdiffusion equations with a time Caputo fractional derivative of order $$\alpha (0<\alpha <1)$$ . We first transform the original equation into a special and equivalent form, which is then discretized by a fourth-order compact finite difference approximation in the spatial directions and by an alternating direction implicit (ADI) approximation in the temporal direction. The resulting compact ADI scheme is uniquely solvable and unconditionally stable. The optimal error estimates in the weighted $$L^{\infty }$$ , $$H^{1}$$ and $$L^{2}$$ norms are obtained, and show that the compact ADI method has the temporal accuracy of order $$\min \{1+\alpha ,2-\alpha \}$$ and the fourth-order spatial accuracy. Applications using three model problems give numerical results that demonstrate the accuracy and the effectiveness of this new method. [ABSTRACT FROM AUTHOR]

We study streamline diffusion schemes applied for numerical solution of the one and one-half dimensional relativistic Vlasov-Maxwell system arising in modeling plasma of particles governed by electro-magnetic fields. We derive stability estimates and prove optimal convergence rates, due to the maximal available regularity of the exact solution. [ABSTRACT FROM AUTHOR]

We construct space-time Petrov-Galerkin discretizations of the heat equation on an unbounded temporal interval, either right-unbounded or left-unbounded. The discrete trial and test spaces are defined using Laguerre polynomials in time and are shown to satisfy the discrete inf-sup condition. Numerical examples are provided. [ABSTRACT FROM AUTHOR]

We propose general spectral and pseudo-spectral Jacobi-Galerkin methods for fractional order integro-differential equations of Volterra type. The fractional derivative is described in the Caputo sense. We provide rigorous error analysis for spectral and pseudo-spectral Jacobi-Galerkin methods, which show that the errors of the approximate solution decay exponentially in $$L^\infty $$ norm and weighted $$L^2$$ -norm. The numerical examples are given to illustrate the theoretical results. [ABSTRACT FROM AUTHOR]

This paper deals with the numerical computation of null controls for the wave equation with a potential. The goal is to compute approximations of controls that drive the solution from a prescribed initial state to zero at a large enough controllability time. In [ Cîndea, Fernández-Cara & Münch, Numerical controllability of the wave equation through primal methods and Carleman estimates, 2013], a so called primal method is described leading to a strongly convergent approximation of boundary controls: the controls minimize quadratic weighted functionals involving both the control and the state and are obtained by solving the corresponding optimality condition. In this work, we adapt the method to approximate the control of minimal square-integrable norm. The optimality conditions of the problem are reformulated as a mixed formulation involving both the state and its adjoint. We prove the well-posedeness of the mixed formulation (in particular the inf-sup condition) then discuss several numerical experiments. The approach covers both the boundary and the inner controllability. For simplicity, we present the approach in the one dimensional case. [ABSTRACT FROM AUTHOR]

We formulate a numerical method to solve the porous medium type equation with fractional diffusion The problem is posed in $$x\in {\mathbb {R}}^N$$ , $$m\ge 1$$ and with nonnegative initial data. The fractional Laplacian is implemented via the so-called Caffarelli-Silvestre extension. We prove existence and uniqueness of the solution of this method and also the convergence to the theoretical solution of the equation. We run numerical experiments on typical initial data as well as a section that summarizes and concludes the proposed method. [ABSTRACT FROM AUTHOR]

We study the stability of some finite difference schemes for symmetric hyperbolic systems in two space dimensions. For the so-called upwind scheme and the Lax-Wendroff scheme with a stabilizer, we show that stability is equivalent to strong stability, meaning that both schemes are either unstable or $$\ell ^2$$-decreasing. These results improve on a series of partial results on strong stability. We also show that, for the Lax-Wendroff scheme without stabilizer, strong stability may not occur no matter how small the CFL parameters are chosen. This partially invalidates some of Turkel's conjectures in Turkel ( 16(2):109-129, ). [ABSTRACT FROM AUTHOR]

We prove in this paper the continuity of the natural projection operator from $W^{1,q}_{0}(\varOmega )^{d}$, q∈[1,+∞), d=2 or d=3, to the MAC discrete space of piecewise constant functions over the dual cells, endowed with the finite volume $W_{0}^{1,q}$-discrete norm. Since this projection operator is also a Fortin operator (that is an operator which 'preserves' the divergence), this result may be applied to control the pressure in mixed problems where the test function for the velocity must be more regular than $H^{1}_{0}(\varOmega )^{d}$. [ABSTRACT FROM AUTHOR]

We consider initial value problems for semilinear parabolic equations, which possess a dispersive term, nonlocal in general. This dispersive term is not necessarily dominated by the dissipative term. In our numerical schemes, the time discretization is done by linearly implicit schemes. More specifically, we discretize the initial value problem by the implicit-explicit Euler scheme and by the two-step implicit-explicit BDF scheme. In this work, we extend the results in Akrivis et al. (Math. Comput. 67:457-477, ; Numer. Math. 82:521-541, ), where the dispersive term (if present) was dominated by the dissipative one and was integrated explicitly. We also derive optimal order error estimates. We provide various physically relevant applications of dispersive-dissipative equations and systems fitting in our abstract framework. [ABSTRACT FROM AUTHOR]

In this paper we present rigorous a posteriori L 2 error bounds for reduced basis approximations of the unsteady viscous Burgers’ equation in one space dimension. The a posteriori error estimator, derived from standard analysis of the error-residual equation, comprises two key ingredients—both of which admit efficient Offline-Online treatment: the first is a sum over timesteps of the square of the dual norm of the residual; the second is an accurate upper bound (computed by the Successive Constraint Method) for the exponential-in-time stability factor. These error bounds serve both Offline for construction of the reduced basis space by a new POD-Greedy procedure and Online for verification of fidelity. The a posteriori error bounds are practicable for final times (measured in convective units) T≈ O(1) and Reynolds numbers ν −1≫1; we present numerical results for a (stationary) steepening front for T=2 and 1≤ ν −1≤200. [ABSTRACT FROM AUTHOR]

The single-step ϑ-scheme on a variable time grid is employed for the approximate solution of the initial-value problem for a nonlinear first-order evolution equation. The evolution equation is supposed to be governed by a possibly time-dependent hemicontinuous operator that is (up to a shift) monotone and coercive, and fulfills a certain growth condition. A piecewise constant as well as piecewise linear prolongation of the time-discrete solution is shown to converge towards the exact solution if ϑ≥1/2 (including the Crank-Nicolson scheme). In the appearance of a strongly continuous perturbation of the monotone main part, the method is still convergent if ϑ>1/2 and if the ratio of adjacent step sizes is bounded from above by a power of ϑ/(1− ϑ). Besides convergence also well-posedness of the time-discrete problem as well as a priori error estimates are studied. [ABSTRACT FROM AUTHOR]

Classical solutions of mixed problems for first-order partial functional differential equations in several independent variables are approximated in this paper by solutions of a difference problem of Euler type. The mesh for the approximate solutions is obtained by the numerical solution of equations of bicharacteristics. Convergence of explicit difference schemes is proved by consistency and stability arguments. It is assumed that the given functions satisfy nonlinear estimates of Perron type. Differential equations with deviated variables and differential integral equations can be obtained from the general model by specifying the given operators. [ABSTRACT FROM AUTHOR]