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Questions tagged [discretization]

The process of representing a continuum space with a finite set of points/elements

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3
votes
1answer
69 views

Why do we have to resort to Higher order schemes for solving the 1-D advection equation/ continuity equation?

\begin{equation} \begin{aligned} \frac{\partial N}{\partial t} &+ \frac{\partial J}{\partial r} = 0, \\ \frac{\partial N}{\partial t} &+ \frac{\partial }{\partial r}(N \upsilon ...
4
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1answer
135 views

Projection method FVM poisson part, adding source term

The idea of the method is to decompose the Navier-Stokes equation into the solenoidal and irrotational parts. $$\frac{\partial u}{\partial t}+u(\nabla \cdot u)=-\frac{1}{\rho}\nabla p+\nabla ^2 u$$ ...
0
votes
1answer
101 views

Discretization of a non-linear ODE using FDM isn't grid indepenent

I am trying to solve the ODE : $\frac{d^2T}{dx^2} = \omega_1 T+\omega_2 T^2$ + using different numerical methods. I have tried the following discretizations so far and none of them seem to be grid ...
1
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0answers
37 views

discretization of advection diffusion with variable coefficients

I am looking for help to find a somewhat stable FD numerical scheme for the advection diffusion equation posed on a curve $(x(r),y(r))$. The equation becomes $$u_t=\alpha(r) u_r +\beta(r) u_{rr}+f(r,t)...
1
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1answer
62 views

Rate of convergence - Stochastic Euler Method

The absolute error criterion of the pathwise approximation of an Ito process $X$ by an Euler approximation $Y$ is: $$ \epsilon=E\left(\left|X_{T}-Y(T)\right|\right) $$ We shall say that a time-...
1
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1answer
81 views

Discretization of a nonlinear boundary value problem

I am trying to use finite element method to discretize the following problem \begin{align} \min_{u \in H^1_0(\Omega)} \int \| \Delta u(x) - 0.5*[u(x) + \langle e, x \rangle + 1]^3 \|^2_2 \ d\Omega, \...
2
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1answer
70 views

FV Discretization of source term in 2D Poisson Equation

I am learning Finite Volume method using the textbook "An Introduction to CFD: Finite Volume Method" by Veersteg and Malalasekera. I would like to solve a heat conduction problem over a ...
2
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0answers
41 views

In sights into why higher order finite differencing method leads faster to instability

I was playing around with numerically solving the 1D wave equation with density and stiffness varying with position using central differencing methods and noticed that for certain discretization steps ...
3
votes
1answer
76 views

Choosing an appropriate time step for a discrete & continuous dynamics simulation

I have inherited of a flight dynamics simulation in C++ which represents a small drone with it's autopilot, actuator dynamics and a solid state IMU. Hence, it is composed of a few models, some ...
0
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0answers
43 views

How to solve for underlying function from discrete data set containing integral of that function

New to Computational Science, I hope I'm on the right exchange network for this question. I have a time series data set that contains the sum of a source data set representing an exponential decay ...
2
votes
3answers
193 views

Flux sign and face normal confusion in finite volume method

I implemented a solver for the 2D steady-state heat equation (without heat generation and homogeneous material) $\nabla. (k\nabla T) = 0$, using finite volume method, however, I am having some ...
3
votes
1answer
143 views

FEM with elastic inhomogeneous properties leads to mesh-induced anisotropy

I'm solving an elastic homogenization problem and I'm having problems with mesh artifacts. I would like to first give a brief summary of what I do: I have a system with inhomogeneous (but isotropic) ...
5
votes
0answers
60 views

Why does the naive barycentric hodgestar fail?

The discrete exterior calculus is defined first using circumcentric dual cells, because the primal and dual edges are orthogonal and thus the dual cells are convex. This leads to a diagonal hodge star ...
0
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0answers
64 views

The relation between PDE order and discretization order

In Jasak's Ph.D. thesis (2000), a notion is given about discretization of a transport equation: For good accuracy, it is necessary for the order of the discretization to be equal to or higher than the ...
6
votes
2answers
255 views

Numerical flux and source term in FVM (Burger's like equation)

I'm trying to solve the following equation with FVM $$u_t + f(u)_x = g(u)$$ where $g$ is some smooth function of $u$ and $f(u) = \frac{u^2}{2}$. This is really similar to Burger's equation, except ...
0
votes
1answer
297 views

Solving differential equation in Python with discretized variable coefficients

I am trying to solve a differential equation with discretized variable coefficients which are calculated from a time serie. In this case the Runge-Kutta step size is fixed by the frequency in the time ...
1
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0answers
60 views

Can the standard multigrid performance be used for time-dependent PDEs?

Consider a time dependent pde(i.e u(x,t)).I know when only space-coarsening is used the standard multigrid performance can be applied but what if instead we use only time-coarsening?Can we apply the ...
4
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1answer
150 views

Differences between Discrete Fourier Transform and Continuous Fourier Transform?

I am trying to visualize the time dependence of a free particle given an initial wave-function using Python and I just wanted to know if I could use the in built FFT implementation from NumPy to find ...
1
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0answers
106 views

FDM discretization of equation on the boundary

In order to simulate the following equation using FDM $$u_t(t,x)-u_{xx}(t,x)=0, \quad (t,x) \in (0,1)\times (0,1)$$ $$(u_t(t,x)-u_{x}(t,x))\rvert_{x=0}=0, \quad t \in (0,1)$$ $$(u_t(t,x)+u_{x}(t,x))\...
1
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1answer
142 views

How to do Weierstrass-transform in MATLAB?

I have a diagonalization problem. I have the eigenstates correctly, and I want to do a Gaussian-smearing (Weierstrass-transform) on them. So I have the wave functions ($\Psi$), and the continuous ...
0
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1answer
86 views

Is “Gradient Computation” in Finite Volume Discretization Really 2nd order accurate?

Based on this, pp 245, we go through these steps to discretize a gradient statement, namely $\nabla\phi$: 1- Gauss theorem reads, $$ \int_V\nabla \phi dV = \oint_{\partial V}\phi dS $$ 2- Integral ...
2
votes
1answer
46 views

Discretization with non-constant matrix containg entries form unknown vector

Consider a system of PDEs $$ \begin{cases} u_t = \nabla \cdot (D(u)\nabla u) + \frac{c}{K_U+c}u-ku\\ c_t = d_c\Delta c -\frac{\nu_U c}{K_U + c}u \end{cases} $$ with some boundary conditions. Here, $D(...
1
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1answer
181 views

Approximation Error in a Finite Difference Approximation of the Square of Derivative

First Part: (First-order derivative) Assuming $f$ is an infinitely differential function everywhere, the Taylor series of $f(x + h)$ at $x$ is \begin{align}\tag{1} f(x + h) = f(x) + hf'(x) + \frac{1}...
0
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0answers
74 views

Analytic vs discrete understanding of PDE

The PDE I am working with: $$\partial_tu = \nabla \cdot (a(x)\nabla u)-\beta(x)u\\ \partial_nu=0, x \in \Omega \subset \mathbb{R}^2\\ \beta(x)>0$$ Integrate the PDE: $$\int_\Omega \partial_t u=\...
0
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0answers
33 views

What will PDE discretization matrix look like for time and space? [duplicate]

Please note: this question is not a duplicate of this question since, while the PDE is the same, the nature of this question is different, i.e. the other question treats a different aspect of this PDE....
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0answers
56 views

Discretizing a parabolic PDE with finite volume method

I want to discretize the following parabolic PDE: $$u_t = \nabla\cdot(\alpha(x)\nabla u)- \beta u\\ x\in\Omega \subset \mathbb{R}^2\\ \partial_n u = 0\\ u(t,0) = u_0(x)\ge 0, \alpha(x)>0$$ Given ...
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0answers
80 views

Determine truncation error of PDE discretization

The equation is $$\frac{\partial}{\partial x}\left(u\frac{\partial u}{\partial x}\right)=f(x)\\ 0<x<1, u(0)=u(1)=0$$ I'm discretizing this PDE using FVM as follows: $0=x_0=x_{1/2}<x_1<x_{...
0
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0answers
53 views

Existence and uniquness of solution of FVM for Poisson equation

I'm discretizing the following Poisson equation using FVM where the domain $\Omega$ of the solution is a regular hexagon of side $1$ centered about the origin. $$\Delta u =k,\text{ $k$ constant}\\ \...
0
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1answer
257 views

Elliptic PDE finite volume method with Dirichlet boundary condition

I want to discretize the following equation using a Finite Volume Method $$\nabla \cdot (a(x)\nabla u)=f(x)\\x\in \Omega \subset \mathbb{R}^2 \\u_{|\partial\Omega}=g$$ I'm using Voronoi cells here: ...
3
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2answers
189 views

Finite volume discretization of non-conservative linear hyperbolic equation

Problem. Consider the one-dimensional adjoint Euler equations for $(x,t) \in \Omega \times [0,T]$ with $\Omega \subset \mathbb{R}$ and $T > 0$ $$ \varphi_t + \Big(\frac{\mathrm{d}F}{\mathrm{d} U}(x)...
1
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0answers
114 views

PDE discretization on triangular domain

Given the 2D Poisson equation $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<y<1\\ \partial_n u (x, 1-x) =0, 0<x<1$$ defined on the domain $\Omega := \{(x,y) \in \...
0
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1answer
856 views

Discretization Neumann boundary condition

I'm currently working with the following Poisson equation with mixed boundary conditions, including a Neumann boundary condition. $$\Delta u = f\\ u(x,0) = g_1(x), 0<x<1\\u(0,y) = g_2(y), 0<...
0
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1answer
73 views

Changing the domain of a 3D Finite Difference code from cube to sphere

I have an explicit FD (Finite Difference) code for diffusion/heat on a PDE in a cuboid domain, and it works fine. I would like to update the discretized equations and change the code so as to solve ...
4
votes
1answer
130 views

Matrix Representation of a Discretization for a Partial Differential Equation

I want to discretize the following problem \begin{cases} \mu \nabla^2u+(\lambda+\mu)\nabla \nabla\cdot u = \rho \frac{\partial^2u }{\partial t^2 } + \beta \frac{\partial u}{\partial t}\\ u(...
6
votes
1answer
117 views

How do I integrate a function defined over an arbitrary area?

Let's say, I have a compact area $S$ (for example a circle, a square or some arbitrary polygon) and a function $f: S \rightarrow \mathbb{R}$. I want to numerically calculate the Integral $$ \int_S f(\...
2
votes
1answer
100 views

Stability Analysis

The partial differential equation, \begin{align} \dfrac{\partial f}{\partial x} + a(x)\dfrac{\partial f}{\partial y} = 0 \qquad & f(0,y) = f(L_1,y) = c_0e^{-y} \\ & f(x,0) = c_0 \;,\; f(x,L_2) ...
5
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1answer
424 views

Finite Differencing schemes for Convection-Diffusion equation

I'm using the Convection(/advection)-Diffusion(-Reaction) equation to calculate the temperature over time in different hydraulic parts like a pipe or a heat exchanger. The flow/convection is always 1D,...
2
votes
1answer
157 views

Is there Von Neumann stability analysis for 9-point laplacian like we have for the 5-point Laplacian?

For spatial accuracy in 2-D Laplace equation, a 9-point stencil is better than a 5-point one. $$\partial_tq= r\left(\partial^2_x q + \partial^2_y q\right)$$ for FTCS (forward-time, central-space) ...
1
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1answer
78 views

How to isolate and test time discretization order of accuracy

I have a code that uses both spatial and time discretization/integration. For convergence analysis, I am wondering how one would test the order of accuracy of their ${time}$ integration scheme? I ...
0
votes
1answer
26 views

Calculating volume of a discretised diffuse interface object

Suppose I have a spherical object projected onto a discrete square mesh. The dicretised circle can be represented by filling a logical matrix such that voxels in the interior of the sphere are filled ...
2
votes
1answer
62 views

Discretisation of logarithmic derivative: Deriving the formula

I'm reading a paper where they use a discrete approximation of a logarithmic mass growth rate as follows: $$ \frac{d \log M}{d \log t} \approx \frac{(t_B + t_A)(M_B - M_A)}{(t_B - t_A)(M_B + M_A)}$$ ...
1
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0answers
52 views

Decrease in slope during convergence analysis

I am using the method of manufactured solutions to perform the order of accuracy testing. I am using a cube for the testing. The cube is size 1m on all sides. I used 5 refinements: $dx = dy = dz = ...
1
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0answers
41 views

Limit to volume change in a discretized mathematical model?

I have set up a mathematical model describing the diffusion of ozone out of a gas bubble. The bubble is surrounded by a thin gas film. So actually, the model describes the diffusion of ozone through ...
2
votes
1answer
116 views

FEM-Laplace with Dirichlet in only a few points: Nonsingular operator?

Let's consider the FEM discretization of the Laplace operator without boundary conditions, i.e., $$ a(u,v) = \int_\Omega \nabla u \cdot \nabla v - \int_\Gamma (n\cdot \nabla u) v. $$ For one-...
5
votes
2answers
351 views

Residual norm of PDE discretization: correspondence in the continuous problem?

Solving a linear PDE like $$ \Delta u = f \quad\text{on } \Omega,\\ n\cdot \nabla u = 0 \quad\text{on } \Gamma, $$ with Finite Elements usually goes like this: Create the discretization $Au=b$ via $$ ...
1
vote
0answers
324 views

Global truncation error behavior at fixed time step

I am trying to solve the following diffusion equation problem: $\frac{\partial f}{\partial t}=\frac{\partial (D\frac{\partial f}{\partial x})}{\partial x}+S$ $D=1+x^{2}+\sin(x)$ $f(x,0)=1 , f(0,t)...
1
vote
3answers
123 views

Discretization Error amplification instead of stagnation to machine precision

I wrote a code on Python 2.7.5 to solve numerically the following differential equation. $\frac{\partial^2f}{\partial x^2}=-S$ $S=\pi^{2}\sin(\pi x)$ S is chosen that way in order to have $f= \sin(\...
1
vote
1answer
56 views

Problems with deriving an equation for a finite-difference scheme given in the journal paper

I'm reading this paper and trying to follow everything that the author has done. A Comparative Study of Finite Volume Method and Finite Difference Method for Convection-Diffusion Problem But there ...
2
votes
0answers
116 views

Monotone, monotonicity preserving, LED, TVD, NVD, bounded, stable and stability preserving discretization schemes [closed]

When it comes to discretization schemes for finite volume method, the following terms can be found in literature: monotone schemes monotonicity preserving schemes local extremum diminishing schemes ...
1
vote
5answers
866 views

Don't we care about the numerical diffusion in the diffusion term?

In the context of the solution of advection-diffusion equations by finite volume method, many numerical schemes, papers and book chapters are dedicated to address the numerical diffusion and/or ...