🔢 mathematics

LDG method for solving spatial and temporal fractional nonlinear convection-diffusion equations

This paper develops a local discontinuous Galerkin (LDG) method using Legendre basis functions to solve nonlinear convection-diffusion equations with space and time-fractional operators, proving its stability and optimal convergence order through numerical analysis.

Original authors: Majid Rajabzadeh, Moein Khalighi

Published 2026-02-11

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Majid Rajabzadeh, Moein Khalighi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how a drop of ink spreads in a glass of water, but there’s a twist: the water isn't just liquid; it’s a strange, "memory-filled" substance. As the ink moves, it doesn't just follow the current; it remembers where it was a moment ago, and the very fabric of the space it moves through is "sticky" and unpredictable.

This paper is about creating a high-tech mathematical "GPS" to track that ink. Here is the breakdown of how they did it.

1. The Problem: The "Memory" and "Sticky Space"

In standard physics, we use equations to predict how things move (convection) and spread out (diffusion). Usually, these equations assume that what happens now only depends on what is happening right now.

However, the authors are looking at Fractional Equations.

Time-Fractional: This means the system has a "memory." The ink's movement now is influenced by its entire history. It’s like trying to drive a car where the steering wheel reacts not just to your current turn, but to every turn you’ve made in the last five minutes.
Space-Fractional: This means the space itself is "jumpy." Instead of moving smoothly from point A to point B, particles might make sudden, long-distance leaps.

Predicting this is incredibly hard because the math becomes "non-linear"—meaning a small change in one spot can cause a massive, unpredictable explosion of change elsewhere.

2. The Solution: The LDG Method (The "Lego" Strategy)

To solve this, the researchers used a method called Local Discontinuous Galerkin (LDG).

Imagine you are trying to map a complex, jagged mountain range. If you try to use one giant, smooth sheet of plastic to cover the whole range, it won't fit the peaks and valleys. Instead, you take thousands of tiny, individual Lego bricks.

Each brick is "discontinuous," meaning they don't have to be perfectly glued together at the edges.
Because they are small and independent, you can snap them together in a way that mimics the jaggedness of the mountains much more accurately.

The "LDG" part is a clever trick where they take one giant, scary, complicated equation and break it down into several smaller, simpler "first-order" equations. It’s like taking a massive, complex IKEA wardrobe and breaking it down into a series of simple, one-step assembly instructions.

3. The Secret Sauce: Legendre Polynomials

The researchers didn't just use any "bricks"; they used special ones called Legendre basis functions.

Think of these as "smart bricks." Most bricks are just blocks, but these are shaped with mathematical precision so that they "fit" together with almost zero wasted space. They are incredibly efficient at smoothing out the errors that usually happen at the edges where two bricks meet. Using these "smart bricks" makes the whole simulation much faster and much more accurate.

4. The Result: Does it work?

The authors didn't just claim it works; they proved it in two ways:

The Proof (Stability): They used heavy-duty math to prove that their method won't "explode." In computer simulations, sometimes a tiny error grows so fast that the whole math model crashes. They proved their "GPS" stays on track.
The Test (Convergence): They ran simulations and compared them to known answers. They found that as they used more "bricks" (making the grid finer), the error dropped exactly as fast as their math predicted. It was like testing a new ruler—the more precise the markings, the more perfect the measurement became.

Summary in a Nutshell

The paper provides a new, highly accurate way to simulate complex, "memory-heavy" movements (like how pollutants spread in groundwater or how heat moves through weird materials) by breaking the world down into tiny, smart, mathematical Lego bricks that can handle the "jumps" and "memories" of the universe.

Technical Summary: LDG Method for Solving Spatial and Temporal Fractional Nonlinear Convection-Diffusion Equations

1. Problem Statement

The paper addresses a complex class of nonlinear fractional partial differential equations (FPDEs) that incorporate both time-fractional and space-fractional operators. The governing equation is a nonlinear convection-diffusion equation featuring:

Time-fractional derivative: A Caputo fractional derivative of order $0 < \alpha \leq 1$ .
Nonlinear Convection: Represented by the term $\frac{\partial F(V)}{\partial \xi}$ , where $F$ is a Lipschitz function.
Nonlinear Diffusion: Represented by $\frac{\partial}{\partial \xi} \left( S(V) \frac{\partial V}{\partial \xi} \right)$ , where $S(V) \geq 0$ is a bounded Lipschitz function.
Fractional Laplacian: A spatial operator $(-\Delta)^{\beta/2}$ of order $1 < \beta < 2$ , defined via a singular integral.

Such equations are critical for modeling phenomena with "memory effects" and long-range dependencies, such as viscoelastic systems, complex networks, and semiconductor device dynamics.

2. Methodology

The authors propose a Local Discontinuous Galerkin (LDG) method to solve this equation. The core technical steps include:

System Reduction: To handle the higher-order fractional derivatives, the original equation is transformed into a system of first-order equations by introducing auxiliary variables ( $E, L, R$ ). This allows the application of the DG framework to higher-order problems.
Basis Functions: The method utilizes Legendre polynomials as the basis functions for the approximation space. The authors argue that the orthogonality and completeness of Legendre polynomials significantly enhance the accuracy and stability of the numerical scheme.
Discretization:
- Temporal Discretization: The time-fractional derivative is approximated using a quadrature-based approach (incorporating weights $W(\alpha_j)$ and step sizes $\Delta t$ ).
- Spatial Discretization: The spatial domain is partitioned into elements, and the LDG scheme is applied using specific numerical fluxes to manage the discontinuities at element interfaces.
Mathematical Rigor: The authors provide a formal proof of stability (showing the $L^2$ norm of the solution remains bounded) and convergence (establishing the optimal error order).

3. Key Contributions

Novel Numerical Scheme: The development of an LDG framework specifically tailored for equations containing both Caputo time-fractional derivatives and fractional Laplacian spatial operators.
Theoretical Guarantees: The paper provides a rigorous mathematical proof that the proposed scheme is stable and achieves an optimal convergence order of:
$\mathcal{O}(h^{k+1} + (\Delta t)^{1+\frac{p}{2}} + p^2)$
where $h$ is the mesh size, $k$ is the polynomial degree, $\Delta t$ is the time step, and $p$ relates to the numerical integration accuracy.
Basis Function Optimization: Demonstrating that the use of Legendre basis functions provides superior performance compared to standard methods.

4. Results and Numerical Validation

The authors validate the method through three distinct numerical examples:

Example 5.1 & 5.2: These test the method on smooth solutions with nonlinear convection and diffusion. The results confirm that as the polynomial degree $k$ increases, the error decreases significantly, matching the predicted convergence rates.
Example 5.3: This tests the method's robustness against discontinuous initial conditions, a common challenge in fractional calculus. The scheme remained stable and accurate.
Convergence Analysis: The numerical tables (Tables 1–4) demonstrate that the error $E_h$ scales predictably with the mesh size $N$ and time step $\Delta t$ , confirming the theoretical error estimates.

5. Significance

This work is significant for the computational mathematics community because it provides a reliable, high-order tool for solving non-local, nonlinear fractional problems. By successfully bridging the gap between the LDG method and fractional Laplacian operators, the paper offers a sophisticated approach for researchers in physics, finance, and biology who require high-precision modeling of anomalous diffusion and memory-dependent processes.