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In this study, we investigate a maximum principle of the Fourier spectral method (FSM) for diffusion equations. It is well known that the FSM is fast, efficient and accurate. The maximum principle holds for diffusion equations: A solution satisfying the diffusion equation has the maximum value under the initial condition or on the boundary points. The same result can hold for the discrete numerical solution by using the FSM when the initial condition is smooth. However, if the initial condition is not smooth, then we may have an oscillatory profile of a continuous representation of the initial condition in the FSM, which can cause a violation of the discrete maximum principle. We demonstrate counterexamples where the numerical solution of the diffusion equation does not satisfy the discrete maximum principle, by presenting computational experiments. Through numerical experiments, we propose the maximum principle for the solution of the diffusion equation by using the FSM.

Abstract In this article, we consider a temporally second-order unconditionally energy stable computational method for the Allen–Cahn (AC) equation with a high-order polynomial free energy potential. By modifying the nonlinear parts in the governing equation, we have a linear convex splitting scheme of the energy for the high-order AC equation. In addition, by combining the linear convex splitting with a strong-stability-preserving implicit–explicit Runge–Kutta (RK) method, the proposed method is linear, temporally second-order accurate, and unconditionally energy stable. Computational tests are performed to demonstrate that the proposed method is accurate, efficient, and energy stable.

In this paper, we introduce a space-fractional version of the molecular beam epitaxy (MBE) model without slope selection to describe super-diffusion in the model. Compared to the classical MBE equation, the spatial discretization is an important issue in the space-fractional MBE equation because of the nonlocal nature of the fractional operator. To approximate the fractional operator, we employ the Fourier spectral method, which gives a full diagonal representation of the fractional operator and achieves spectral convergence regardless of the fractional power. And, to combine with the Fourier spectral method directly, we present a linear, energy stable, and second-order method. Then, it is possible to simulate the dynamics of the space-fractional MBE equation efficiently and accurately. By using the numerical method, we investigate the effect of the fractional power in the space-fractional MBE equation.

In this study, we present a classification method for ternary small data using the modified ternary Allen–Cahn (tAC) system. The governing system is the tAC equation with the fidelity term, which keeps the solution as close as possible to the given data. To solve the tAC system with the fidelity term, we apply an operator splitting method. We use an implicit-explicit finite difference method for solving the split equations. To validate the robust and superior performance of the proposed numerical algorithm, we perform the comparison tests with other widely used classifiers such as logistic regression, decision tree, support vector machine, random forest, and artificial neural network for small datasets.

In this paper, we introduce a new fractional-in-space modified phase-field crystal equation based on the L2-gradient flow approach, where the mass of atoms is conserved by using a nonlocal Lagrange multiplier. To solve the L2-gradient flow-based fractional-in-space modified phase-field crystal equation, we present a mass conservative and energy stable method based on the convex splitting idea. Numerical examples together with standard tests in the classical H−1-gradient flow-based modified phase-field crystal equation are provided to illustrate the applicability of the proposed framework.

In this paper, we present benchmark problems for the numerical discretization of the Cahn–Hilliard equation with a source term. If the source term includes an isotropic growth term, then initially circular and spherical shapes should grow with their original shapes. However, there is numerical anisotropic error and this error results in anisotropic evolutions. Therefore, it is essential to use isotropic space discretization in the simulation of growth phenomenon such as tumor growth. To test numerical discretization, we present two benchmark problems: one is the growth of a disk or a sphere and the other is the growth of a rotated ellipse or a rotated ellipsoid. The computational results show that the standard discrete Laplace operator has severe grid orientation dependence. However, the isotropic discrete Laplace operator generates good results.

In this study, we present an unconditionally stable positivity-preserving numerical method for the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation in the one-dimensional space. The Fisher–KPP equation is a reaction-diffusion system that can be used to model population growth and wave propagation. The proposed method is based on the operator splitting method and an interpolation method. We perform several characteristic numerical experiments. The computational results demonstrate the unconditional stability, boundedness, and positivity-preserving properties of the proposed scheme.

To solve the L2-gradient flow-based phase-field crystal equation accurately and efficiently, we present a linear, second-order, and unconditionally energy-stable method. We first truncate the quartic function in the Swift–Hohenberg energy functional. We also put the truncated function in the expansive part of the energy and add an extra term to have a linear convex splitting. Then, we apply the linear convex splitting to both the L2-gradient flow and the nonlocal Lagrange multiplier terms and combine it with the second-order SSP-IMEX-RK method. We prove that the proposed method is mass-conservative and unconditionally energy-stable. Numerical experiments including standard tests in the classical H−1-gradient flow-based phase-field crystal equation support that the proposed method is second-order accurate in time, mass conservative, and unconditionally energy-stable.

The conservative Swift–Hohenberg equation was introduced to reformulate the phase-field crystal model. A challenge in solving the conservative Swift–Hohenberg equation numerically is how to treat the nonlinear term to preserve mass conservation without compromising efficiency and accuracy. To resolve this problem, we present a linear, high-order, and mass conservative method by placing the linear and nonlinear terms in the implicit and explicit parts, respectively, and employing the implicit-explicit Runge–Kutta method. We show analytically that the method inherits the mass conservation. Numerical experiments are presented demonstrating the efficiency and accuracy of the proposed method. In particular, long time simulation for pattern formation in 2D is carried out, where the phase diagram can be observed clearly. The MATLAB code for numerical implementation of the proposed method is provided in Appendix.

In this paper, we review the Fourier-spectral method for some phase-field models: Allen–Cahn (AC), Cahn–Hilliard (CH), Swift–Hohenberg (SH), phase-field crystal (PFC), and molecular beam epitaxy (MBE) growth. These equations are very important parabolic partial differential equations and are applicable to many interesting scientific problems. The AC equation is a reaction-diffusion equation modeling anti-phase domain coarsening dynamics. The CH equation models phase segregation of binary mixtures. The SH equation is a popular model for generating patterns in spatially extended dissipative systems. A classical PFC model is originally derived to investigate the dynamics of atomic-scale crystal growth. An isotropic symmetry MBE growth model is originally devised as a method for directly growing high purity epitaxial thin film of molecular beams evaporating on a heated substrate. The Fourier-spectral method is highly accurate and simple to implement. We present a detailed description of the method and explain its connection to MATLAB usage so that the interested readers can use the Fourier-spectral method for their research needs without difficulties. Several standard computational tests are done to demonstrate the performance of the method. Furthermore, we provide the MATLAB codes implementation in the Appendix A.

In this paper, we reformulate the diffuse interface model of the tumor growth (S.M. Wise et al.,Three-dimensional multispecies nonlinear tumor growth-I: model and numerical method, J. Theor.Biol. 253 (2008) 524--543). In the new proposed model, we use the conservative second-orderAllen--Cahn equation with a space--time dependent Lagrange multiplier instead of using thefourth-order Cahn--Hilliard equation in the original model. To numerically solve the new model, weapply a recently developed hybrid numerical method. We perform various numerical experiments. Thecomputational results demonstrate that the new model is not only fast but also has a good featuresuch as distributing excess mass from the inside of tumor to its boundary regions.

Tissue growth on bioscaffolds can be controlled using substrate geometry such as substrate curvature. In this study, we present a mathematical model and numerical simulation method for tissue growth on a bioscaffold to investigate the effect of local curvature on tissue growth. The mathematical model is based on the Allen−Cahn (AC) equation, which has been extensively used to model many problems involving motion by mean curvature. By solving the AC equation using the explicit Euler method, the proposed method is simple and fast. Numerical simulations on various geometries are presented to demonstrate the applicability of the proposed framework on tissue growth on a bioscaffold.

Oncolytic viruses are genetically engineered viruses that are designed to kill cancer cells while doing minimal damage to normal healthy tissue. After being injected into a tumor, they infect cancer cells, multiply inside them, and when a cancer cell is killed they move on to spread and infect other cancer cells. Chondroitinase ABC (Chase-ABC) is a bacterial enzyme that can remove a major glioma ECM component, chondroitin sulfate glycosoamino glycans from proteoglycans without any deleterious effects in vivo. It has been shown that Chase-ABC treatment is able to promote the spread of the viruses, increasing the efficacy of the viral treatment. In this paper we develop a mathematical model to investigate the effect of the Chase-ABC on the treatment of glioma by oncolytic viruses (OV). We show that the model's predictions agree with experimental results for a spherical glioma. We then use the model to test various treatment options in the heterogeneous microenvironment of the brain. The model predicts that separate injections of OV, one into the center of the tumor and another outside the tumor will result in better outcome than if the total injection is outside the tumor. In particular, the injection of the ECM-degrading enzyme (Chase-ABC) on the periphery of the main tumor core need to be administered in an optimal strategy in order to infect and eradicate the infiltrating glioma cells outside the tumor core in addition to proliferative cells in the bulk of tumor core. The model also predicts that the size of tumor satellites and distance between the primary tumor and multifocal/satellite lesions may be an important factor for the efficacy of the viral therapy with Chase treatment.