Three Core Methods for Numerical Solutions of Partial Differential Equations: From Finite Differences to Finite Elements

# 1. Overview of Numerical Methods for Partial Differential Equations

Partial differential equations (PDEs) are mathematical equations that describe how continuous variables change over time and space in physical phenomena. Due to the general intractability of PDEs for analytical solutions, numerical methods are employed to obtain approximate solutions.

The fundamental principle of numerical methods for PDEs involves discretizing the PDE into a system of algebraic equations, ***mon numerical methods include the finite difference method, finite element method, and finite volume method. These methods each have their own strengths and weaknesses and are suitable for different types of PDEs and application areas.

In this chapter, we will introduce the basic concepts, commonly used methods, and examples of practical applications of numerical methods for PDEs.

# 2. Finite Difference Method

The finite difference method is a numerical technique used to solve partial differential equations (PDEs). It is based on the discretization of PDEs into a set of algebraic equations, which are then solved to obtain approximate solutions.

### 2.1 Basic Principles of the Finite Difference Method

#### 2.1.1 Spatio-temporal Discretization

The finite difference method first discretizes the PDE in the spatio-temporal domain. Spatio-temporal discretization involves dividing the continuous spatio-temporal domain into discrete grid points and defining the values of the unknown function at these grid points.

For a one-dimensional PDE, spatio-temporal discretization can be represented as:

∂u/∂t ≈ (u_i^(n+1) - u_i^n) / Δt
∂u/∂x ≈ (u_i+1^n - u_i^n) / Δx

where u_i^n represents the value of the unknown function at grid point (i, n), and Δt and Δx represent the time step and spatial step, respectively.

#### 2.1.2 Construction of Difference Schemes

After discretization, difference schemes need to be constructed to approximate the derivative terms of the PDE. There are various types of difference schemes, with common ones including:

- **Forward difference scheme:**

∂u/∂x ≈ (u_i+1^n - u_i^n) / Δx

- **Backward difference scheme:**

∂u/∂x ≈ (u_i^n - u_i-1^n) / Δx

- **Central difference scheme:**

∂u/∂x ≈ (u_i+1^n - u_i-1^n) / (2Δx)

Different difference schemes have different levels of accuracy and stability, and the appropriate scheme should be chosen based on the specific problem.

### 2.2 Solution Methods for the Finite Difference Method

There are several methods to solve a system of finite difference equations, including:

#### 2.2.1 Direct Solution Method

The direct solution method converts the system of finite difference equations into a system of linear equations, which are then solved using Gaussian elimination or other direct solution methods.

#### 2.2.2 I***

***mon iterative solution methods include:

- **Jacobi Iteration Method:**

u_i^(n+1) = (b_i - ∑_{j≠i} a_{ij} u_j^n) / a_{ii}

- **Gauss-Seidel Iteration Method:**

u_i^(n+1) = (b_i - ∑_{j<i} a_{ij} u_j^(n+1) - ∑_{j>i} a_{ij} u_j^n) / a_{ii}

- **Successive Over-Relaxation Method:**

u_i^(n+1) = (1 - ω) u_i^n + ω (b_i - ∑_{j≠i} a_{ij} u_j^(n+1)) / a_{ii}

where ω is the relaxation factor, with a range of (0, 2).

Different iterative solution methods have varying convergence rates and stability, and the appropriate method should be chosen based on the specific problem.

# 3.1 Basic Principles of the Finite Element Method

The finite element method is a numerical technique that discretizes continuous problems into discrete ones. The fundamental principle involves dividing the solution domain into a finite number of elements and approximating the unknown solution using a finite number of basis functions.

#### 3.1.1 Establishment of the Weak Form

The weak form is an integral form of the partial differential equation, which can transform the PDE into an equivalent integral equation. For a second-order elliptic PDE, the weak form is:

∫Ω(∇u⋅∇v-fu)dΩ=0

where u is the unknown solution, v is any smooth function, Ω is the solution domain, and f is a known function.

#### 3.1.2 Construction of the Finite Element Space

The finite element space is a function space spanned by basis functions. Different methods can be used to construct the finite element space for a given solution domain, with common methods being:

***Lagrange finite elements:** Basis functions are locally supported continuous functions, polynomials within the element.
***Hermite finite elements:** Basis functions are locally supported discontinuous functions, polynomials within the element, and satisfy continuity conditions at the element boundaries.

### 3.2 Solution Methods for the Finite Element Method

The steps to solve a PDE using the finite element method are as follows:

1. **Discretization:** Divide the solution domain into a finite number of elements and construct the finite element space.
2. **Basis Function Selection:** Select basis functions from the finite element space and use them to approximate the unknown solution.
3. **Galerkin Method:** Substitute the weak form into the basis functions and integrate, resulting in a discrete system of equations.
4. **Petrov-Galerkin Method:** Based on the Galerkin method, introduce weight functions to obtain a weighted discrete system of equations.

#### 3.2.1 Galerkin Method

The Galerkin method is one of the most commonly used finite element solution methods. The basic idea is to substitute the weak form into the basis functions and integrate to obtain a discrete system of equations:

∫Ω(∇u⋅∇v-fu)dΩ=0,   ∀v∈V

where V is the finite element space.

#### 3.2.2 Petrov-Galerkin Method

The Petrov-Galerkin method is an improved version of the Galerkin method. The basic idea is to introduce a weight function w, resulting in a weighted discrete system of equations:

∫Ω(∇u⋅∇w-fw)dΩ=0,   ∀w∈W

where W is the weight function space.

# 4. Finite Volume Method

### 4.1 Basic Principles of the Finite Volume Method

#### 4.1.1 Division of Control Volumes

The finite volume method divides the computational domain into a series of non-overlapping control volumes, each representing a small part of the computational domain. The shape of the control volume can be any polygon, but rectangles or triangles are commonly used.

#### 4.1.2 Discretization of Conservation Laws

The fundamental principle of the finite volume method is based on the discretization of conservation laws. For a control volume, the conservation law states that the net inflow rate of a physical quantity within the control volume is equal to the rate of change of that physical quantity within the control volume.

For a control volume, the conservation law can be represented as:

\frac{\partial}{\partial t} \int_{\Omega} u d\Omega + \int_{\partial \Omega} F(u) \cdot n dS = 0

where:

* $\Omega$ is the control volume
* $u$ is the physical quantity
* $F(u)$ is the flux of the physical quantity
* $n$ is the outward normal vector on the control volume boundary

After discretization, the conservation law becomes:

\frac{\partial u_i}{\partial t} V_i + \sum_{j=1}^{N_f} F_{i,j} = 0

where:

* $u_i$ is the average value of the physical quantity within control volume $\Omega_i$
* $V_i$ is the volume of control volume $\Omega_i$
* $N_f$ is the number of faces of control volume $\Omega_i$
* $F_{i,j}$ is the flux between control volumes $\Omega_i$ and $\Omega_j$

### 4.2 Solution Methods for the Finite Volume Method

#### 4.2.1 Solution Methods for the Finite Volume Method

The solution methods for the finite volume method typically involve the following steps:

1. **Spatial Discretization:** Divide the computational domain into a series of control volumes.
2. **Discretization of Conservation Laws:** Discretize the conservation laws into a set of algebraic equations on control volumes.
3. **Time Discretization:** Discretize the algebraic equations into a sequence over time.
4. **Solving the Algebraic Equations:** Solve the algebraic equations at each time step.

#### 4.2.2 Calculation of Numerical Fluxes

Numerical fluxes are approximate values of the physical quantity at the boundaries of control volumes. There are various methods for calculating numerical fluxes, with common ones including:

***Central Difference Scheme:** Approximate the flux as the physical quantity at the center of the control volume.
***Upwind Scheme:** Approximate the flux as the physical quantity from the upstream control volume.
***Hybrid Scheme:** Approximate the flux as a weighted average of the central difference scheme and upwind scheme.

# 5.1 Applications in Fluid Mechanics

Numerical methods for solving partial differential equations have a wide range of applications in fluid mechanics, with the most representative being the solution of the Navier-Stokes equations. The Navier-Stokes equations are a set of nonlinear partial differential equations that describe the motion of viscous fluids.

### 5.1.1 Solving the Navier-Stokes Equations

There are many numerical methods for solving the Navier-Stokes equations, among which the finite difference method, finite element method, and finite volume method are the most commonly used.

**Finite Difference Method**

The finite difference method discretizes the fluid domain into a grid of cells and uses difference schemes to transform the PDE into a system of algebraic equations. These algebraic equations can be solved using direct solution methods or iterative solution methods.

**Finite Element Method**

The finite element method discretizes the fluid domain into a series of finite elements and constructs a weak form equation using the finite element space. The weak form equation can be solved using the Galerkin method or the Petrov-Galerkin method.

**Finite Volume Method**

The finite volume method discretizes the fluid domain into control volumes and uses conservation laws to discretize the control volumes. The discretized conservation law equations can be solved using finite volume method solution methods.

### 5.1.2 Flow Visualization

Numerical methods for solving partial differential equations can also be used for flow visualization. By post-processing flow field data, visualization results such as streamlines and contours can be generated, intuitively showing the motion of fluids.

For example, the following diagram shows the results of flow visualization in a two-dimensional fluid field obtained by solving the Navier-Stokes equations using the finite volume method.

graph LR
subgraph Navier-Stokes Equations
     A[Navier-Stokes Equations] --> B[Finite Difference Method]
     A[Navier-Stokes Equations] --> C[Finite Element Method]
     A[Navier-Stokes Equations] --> D[Finite Volume Method]
end
subgraph Flow Visualization
     B[Finite Difference Method] --> E[Streamlines]
     C[Finite Element Method] --> F[Contours]
     D[Finite Volume Method] --> G[Flow Field Visualization]
end