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Suppose I had the following periodic 1D advection problem:

$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a jump discontinuity at $x^*\in (0,1)$.

It is my understanding that for a first order upwind scheme (and other low order finite difference schemes) linear finite difference schemes of higher than first order, spurious oscillations occur near the discontinuity as it is advected over time, resulting in a distortion of the solution from its expected wave shape. According to wikipedia explanation, it seems that these oscillations typically occur when a discontinuous function is approximated with a finite fourier series.

For some reason, I can't seem to grasp how a finite fourier series can be observed in the solution of this PDE. In particular, how can I estimate a bound on the "over-shoot" analytically?

Suppose I had the following periodic 1D advection problem:

$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a jump discontinuity at $x^*\in (0,1)$.

It is my understanding that for linear finite difference schemes of higher than first order, spurious oscillations occur near the discontinuity as it is advected over time, resulting in a distortion of the solution from its expected wave shape. According to wikipedia explanation, it seems that these oscillations typically occur when a discontinuous function is approximated with a finite fourier series.

For some reason, I can't seem to grasp how a finite fourier series can be observed in the solution of this PDE. In particular, how can I estimate a bound on the "over-shoot" analytically?

Suppose I had the following periodic 1D advection problem:

$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a jump discontinuity at $x^*\in (0,1)$.

It is my understanding that for a first order upwind scheme (and other low order finite difference schemes), spurious oscillations occur near the discontinuity as it is advected over time, resulting in a distortion of the solution from its expected wave shape. According to wikipedia explanation, it seems that these oscillations typically occur when a discontinuous function is approximated with a finite fourier series.

For some reason, I can't seem to grasp how a finite fourier series can be observed in the solution of this PDE. In particular, how can I estimate a bound on the "over-shoot" analytically?

Suppose I had the following periodic 1D advection problem:

$\frac{\partial u}{\partial t} + c\frac{\partial u}{\partial x} = 0$ in $\Omega=[0,1]$
$u(0,t)=u(1,t)$
$u(x,0)=g(x)$
where $g(x)$ has a jump discontinuity at $x^*\in (0,1)$.

It is my understanding that for a first order upwind scheme (and other low order finite difference schemes) linear finite difference schemes of higher than first order, spurious oscillations occur near the discontinuity as it is advected over time, resulting in a distortion of the solution from its expected wave shape. According to wikipedia explanation, it seems that these oscillations typically occur when a discontinuous function is approximated with a finite fourier series.

For some reason, I can't seem to grasp how a finite fourier series can be observed in the solution of this PDE. In particular, how can I estimate a bound on the "over-shoot" analytically?