High‐order characteristic‐tracking strategy for simulation of a nonlinear advection–diffusion equation

Abstract

In this study, new high‐order backward semi‐Lagrangian methods are developed to solve nonlinear advection–diffusion type problems, which are realized using high‐order characteristic‐tracking strategies. The proposed characteristic‐tracking strategies are second‐order L‐stable and third‐order L(α)‐stable methods, which are based on a classical implicit multistep method combined with a error‐correction method. We also use backward differentiation formulas and the fourth‐order finite‐difference scheme for diffusion problem discretization in the temporal and spatial domains, respectively. To demonstrate the adaptability and efficiency of these time‐discretization strategies, we apply these methods to nonlinear advection–diffusion type problems such as the viscous Burgers' equation. Through simulations, not only the temporal and spatial accuracies are numerically evaluated but also the proposed methods are shown to be superior to the compared existing characteristic‐tracking methods under the same rates of convergence in terms of accuracy and efficiency. Finally, we have shown that the proposed method well preserves the energy and mass when the viscosity coefficient becomes zero.

Simulation of advection–diffusion–dispersion equations based on a composite time discretization scheme

“…where S(z) = 12-18z+11z 2 -3z 3 Remark that μ m+1 p (p = 1, 2) in (11) or (12) does not affect the linear stability of the proposed departure traceback scheme, since u(t, x * (t)) = λx * (t) and its Jacobian is λ. As shown numerically in Fig.…”

“…Among them, the backward semi-Lagrangian method (BSL) describes the movement of particles in the Lagrangian description and resets the positions of the particles to the Eulerian grid point at each time level to reduce the accumulation and collision of particles. Moreover, it is well known that BSL exhibits good stability, allowing the use of larger temporal steps than the spatial grid size by solving equations implicitly along characteristic curves of particles in the reverse direction to time evolution [3,5,30,31]. Therefore, this paper proposes a robust high-order BSL algorithm for solving nonlinear advection-diffusion-dispersion equations.…”

“…The remainder of this paper is structured as follows. Section 2 provides temporal and spatial semi-discretization systems to solve a diffusion-dispersion problem (3). Section 3 designs the departure traceback method to find particle traces along characteristic curves in the reverse time direction by solving the nonlinear initial value problem (IVP) and discusses the stability of the proposed method.…”

Higher‐order explicit schemes based on the method of characteristics for hyperbolic equations with crossing straight‐line characteristics

“…An extensive list of physical applications in linear and nonlinear optics, plasma physics, atomic physics, and relativistic field theory where such models arise is found in Section 2 of [1] (Refs. [4,5,[20][21][22][23][24][25][26][27][28][29]44-64] there); see also Refs. [2][3][4][5][6][7][8][9][10][11][12][13] here and Ref.…”

Reinterpretation of Multi-Stage Methods for Stiff Systems: A Comprehensive Review on Current Perspectives and Recommendations

“…Considering the impact, a new perspective to compare the potentials of both methods should be investigated as well as existing comparative studies. First of all, it is well known that the highest order of an A-stable multi-step method is two, so lots of research [12][13][14][15][16][17][18][19][20][21][22][23][24] developing higher order methods have focused on either multi-step methods satisfying some less restrictive stability condition or multi-stage methods which combine A-stability with high-order accuracy [2,[25][26][27][28][29]. In addition, multi-stage methods such as Runge-Kutta (RK) type methods do not require any additional memory for function values at previous steps since it does not use any previously computed values [30][31][32].…”